The Element-Free Galerkin (EFG) method was originally developed for linear solid mechanics problems, using Moving Least Squares (MLS) approximations to construct shape functions for the numerical approximation of the displacement field and its variations within the weak form of the equilibrium equations. Over the past decades, it has evolved into a versatile meshfree framework applicable to a broad spectrum of engineering and scientific problems. This review provides a comprehensive account of the main advances in EFG, tracing its development from the original formulation and early challenges to the strategies devised to overcome them. Subsequent improvements in accuracy, stability, and computational efficiency are examined in detail, together with alternative shape function constructions such as Moving Kriging (MK) and Local Maximum Entropy (LME) approximations. The extension of EFG to multiphysics problems is discussed, emphasizing how analogies with the Finite Element Method (FEM) have enabled the adaptation of established stabilization and enrichment techniques. Hybrid FEM–EFG coupling strategies are also reviewed. The article concludes with a survey of significant applications in mechanics and transport phenomena, highlighting their broader implications in science and technology.
Element-free galerkinshape functionsapplicationsmultiphysicsFEM-EFG couplingmeshfree methodsSanta Fe Agency for Science, Technology and Innovation (ASACTEI)IO-2025-428Introduction
Recent advances in numerical methods have enabled the solution of a wide range of problems arising in science and engineering, leading to significant developments in computational mechanics and physics. Mesh-based methods such as the Finite Element Method (FEM) [1] and the Finite Volume Method (FVM) [2] remain the most widely employed techniques for the numerical approximation of governing equations, and their capabilities have been demonstrated in increasingly complex applications. These include linear and non-linear problems in solid mechanics [3], fluid dynamics [2,4], thermo-mechanics [3] and multiphysics [5,6], and even topology optimisation [7]. Such mesh-based numerical approaches rely on piecewise approximations of the field variables, which often pose challenges when capturing steep gradients [8–11] or ensuring the smoothness of derivatives [12–15]. Such a feature makes it difficult to handle problems involving large geometric deformations [16,17], moving boundaries [8,10,11], material discontinuities [11,18,19], or evolving interfaces that seldom align with the element or cell boundaries [11,15,20]. Although these issues can be overcome via adaptive re-meshing techniques, their implementation in complex geometries is computationally demanding and requires mapping of field variables between successive meshes. This process introduces additional costs and may degrade the accuracy and stability of the numerical solution [15,20,21]. To overcome these limitations, alternative strategies have been proposed and successfully applied. These include advanced mesh-handling techniques—such as arbitrary Lagrangian–Eulerian (ALE) formulations [22], sliding [23] and overlapping [24,25] meshes, immersed boundary methods [26], level set techniques [27,28], and also approaches that relax the dependence on mesh quality such as smoothed FEM variants [29] and meshfree methods [12]. Within the latter, the Element-Free Galerkin (EFG) method has emerged as a particularly powerful framework, offering flexibility in the discretisation of complex domains and the treatment of problems that challenge traditional mesh-based schemes. EFG methods were formally introduced by Belytschko et al. in 1994 [30] as an alternative framework for the numerical solution of the Galerkin weak form of the conservation equations, employing shape functions constructed from moving least squares (MLS) approximations. The method was used in the solution of benchmark elliptical-linear problems concerning elasticity, fracture mechanics and steady-state heat conduction. This pioneering work put into clear perspective several aspects concerning the accuracy and implementation of this then-emerging numerical approach, particularly the imposition of essential boundary conditions and the numerical integration requirements for improved accuracy. Since MLS approximations do not satisfy the Kronecker delta property, the use of Lagrange multipliers was proposed as the first strategy for enforcing essential boundary conditions. Furthermore, it was demonstrated that remarkable accuracy can be achieved through the use of high-order quadrature rules for numerical integration. Another major advantage highlighted was the absence of any post-processing requirement for the achievement of smooth physical fields obtained as derivatives of the primary variables, unlike in standard mesh-based techniques such as FEM. Subsequently, weighted orthogonal basis functions were introduced to construct MLS approximations at each quadrature point without the need to solve a system of linear equations [31]. This procedure was later termed improved MLS (IMLS), and the corresponding Galerkin weak formulation gave rise to the improved EFG (IEFG) method [32]. The first developments on EFG methods reflected an early recognition of their potential for practical adoption, mostly in elasticity problems. Within this framework, the intrinsic features of EFG enabled the design of simple and effective techniques to address classical challenges such as the capture of stress concentrations [30,31], material discontinuities [33], and the prevention of locking phenomena in nearly incompressible materials [30,34] and bending-dominated problems [31,35]. Although issues such as volumetric and shear locking may still arise in EFG formulations, these effects tend to be less severe compared to their FEM counterparts [30,31]. The growing dissemination, increasing recognition, and continuous refinement of EFG methods motivated works devoted to facilitating their numerical implementation, such as the development of dedicated libraries for the computation of MLS approximations [36] and also introductory contributions aimed at guiding researchers in programming the method [37]. Further efforts to improve the computational efficiency of EFG methods enabled the first implementations for three-dimensional problems [38], consolidating this mesh-less approach as a versatile framework for computational mechanics. As EFG methods have been applied to progressively larger and more complex problems, considerable and continuing efforts have been devoted to mitigating their computational challenges. The construction of shape functions for EFG methods involves the search for neighbouring nodes supporting each integration point in the problem domain, which is actually the most time-consuming stage in the computational implementation of EFG methods [39–42]. Moreover, both sparse matrix assembly and shape function construction become increasingly demanding as the number of supporting nodes per integration point grows [43]. This is mainly because achieving stable results and optimal convergence typically requires the use of high-order integration rules [44,45]. The approaches proposed to enhance computational performance in EFG methods include: the development of more efficient integration schemes such as post-processing techniques for nodal integrations based on Voronoi diagrams [46,47], virtual element decompositions [48,49], stabilised [50] and variationally consistent [51–53] formulations, reproducing kernel gradient smoothing frameworks [54–56] and also projection of EFG shape functions in FEM spaces [57]; improved implementation of sparse matrices assembly procedures and construction of basis functions [43]; computation of deferred correction vectors for the solution of non-linear problems as alternative to matrix reassembly [11]; application of reduced order modelling techniques [58–60]; more efficient neighbouring nodes searching procedures [40,43]; construction of approximations in complex variable spaces to reduce the dimensionality of basis functions in both MLS [61,62] and IMLS [63,64] approximations; and the implementation of techniques to reduce the number of nodes required to represent the problem domain, such as dimension splitting techniques for complex variable [65,66] and standard MLS [62,63,67], and overlapping discretisation approaches [19,41,42]. The ongoing development of procedures devote to improving efficiency, accuracy, and stability of EFG methods has enabled their extension to increasingly complex frameworks in computational mechanics, beyond linear elasticity and other standard elliptical problems. Some of these include nonlinear heat transfer with phase change involving transient conduction in fixed [68–71] and variable domains with moving boundaries [8,10,72], moving heat sources [41,42] and also orthotropic media [20], advection–diffusion [9,11,73,74] mechanisms, linear [18,75–77] and non-linear [16,78–80] thermo-mechanics, plasticity [16,81,82], elastoplasticity [83–85], large deformation for incompressible [86–89] and near-compressible [17,90–92] hyper-elasticity and plasticity [16,93], anisotropy [18,20,94–96], analysis of incompressible Stokes flow under two-level [97,98], generalised [67,99,100] and stabilised divergence-free [66,101] formulations, non-linear fluid-dynamics problems that include implementation of standard Characteristic Based Split techniques for transient analysis [102–104] within EFG frameworks, mixed u-p formulations [105,106], fluid-structure interaction [15], benchmark analyses at high Reynolds numbers [107,108], and free surface flow [109], coupled heat transfer and fluid flow [11,110–113], viscoelasticity [114] and viscoelastic flows [115], linear [116] and non-linear [112,117,118] porous media flow problems, other multiphysics scenarios such as magnetohydrodynamics for both moderate [60,119,120] and high Hartmann numbers [121–124] and phase field modelling in fracture mechanics [125–128], heat transfer [129–133], structural [133–137] and thermo-mechanical [137–140] topology optimisation, and even the solution of partial differential equations in modern physics [141–144] and biological [145–147] applications.
The studies reported so far put into an appropriate perspective how the various improvements and modifications introduced in EFG formulations enabled their widespread application beyond the linear elasticity framework within which they were originally developed. This proliferation has also been accelerated by the fact that most of the enhancements and adaptations implemented to extend EFG methods to transport phenomena and multiphysics problems have been largely based on its analogies with FEM, beyond the well-known differences concerning the construction of shape functions and the assembly of the algebraic system of equations. Consequently, most of the adaptations and improvements of EFG have been achieved through the incorporation of techniques already well established and extensively developed within the FEM framework. This has been achieved by retaining the distinctive advantages of EFG regarding (i) the possibility of more easily achieving higher-order approximations with continuous derivatives than FEM and other mesh-based techniques, and (ii) the greater flexibility in adding or removing nodes [12,19,42]. In contrast, achieving such continuity in FEM is challenging and typically requires either the use of highly complex finite elements with numerous nodal unknowns or post-processing to handle the discontinuous derivative fields generated by simpler elements [12,19,42,148]. These particular features have strongly encouraged the implementation of EFG methods for the solution of increasingly complex problems beyond standard linear elasticity and elliptic potential cases. The techniques inherited from FEM to address difficulties common to this well-established mesh-based framework include the implementation of dimension-splitting methods (DSM) to transform higher-dimensional heat transfer [62], fluid dynamics [67], advection-diffusion [149–151] or even wave propagation [63] problems into multiple lower-dimensional sub-problems sequentially coupled via finite differences; proper orthogonal decomposition (POD) to construct reduced-order models (ROM) with fewer degrees of freedom [58–60]; variational multiscale techniques to stabilise saddle-point problems arising in multiphysics from the coupling of fields such as velocity-pressure in linear [97,98] a non-linear [60,108,109,152] fluid dynamics, electric potential-current density-axial velocity for fully developed MHD flow in channels [119,123,124], temperature-pressure-velocity in coupled fluid flow heat transfer [110–112,153], and velocity-pressure coupling in porous media flow problems [116,118], and to suppress spurious oscillations under advection-dominated conditions in pure convection-diffusion [58,149,154–156], convection-diffusion-reaction [157–159], discontinuities capturing [160] and fluid dynamics [108,152]; reduced integration schemes to mitigate chequerboard-type instabilities in incompressible flow problems [15,106,107] and heat transfer coupling [11,113]; extended approximations with cover functions based on linear combinations of polynomials to satisfy the inf–sup condition in Stokes flow problems [67,99]; streamline-upwind Petrov–Galerkin (SUPG) stabilisation for advection-dominated regimes [9,11,15,42,113]; adaptive refinement strategies; and the implementation of immersed boundary techniques [15]. These developments have given rise to EFG-based adapted formulations such as the Variational Multiscale EFG (VMEFG) [109,111,119], the Generalised EFG (GEFG) [67,99], the Reduced Integration Penalty EFG (RIP-EFG) [11,15,106,107], the SUPG-stabilised EFG [9,11,15,42], and the Penalty-Based Immersed Boundary EFG (PBIB-EFG) [15].
The analogies between FEM and EFG formulations have also fostered the emergence of hybrid approaches that combine the advantages of both methods within a unified computational framework. Since most EFG formulations employ shape functions that do not satisfy the Kronecker delta property, the direct imposition of essential (Dirichlet-type) boundary conditions and the coupling with standard FEM approximations require special attention. Hybrid FEM–EFG techniques were therefore developed to exploit the flexibility and higher accuracy of EFG in regions demanding fine resolution, while retaining the computational efficiency of FEM elsewhere. Early coupling strategies, such as the non-overlapping procedure proposed by Belytschko et al. [161], used interface elements to bridge displacement fields and compensate for the lack of Kronecker delta property in MLS approximations. Although these methods did not ensure perfect continuity of shape function derivatives across the interface, the effect on overall accuracy was limited as the coupling affected only a small portion of the domain. Subsequent developments sought to remove the need for explicit interface elements, including hierarchical IEFG–FEM blending schemes [162], Lagrange multiplier-based couplings [163], and integration constraint formulations [164]. More recent advances have achieved seamless direct coupling by exploiting shape functions satisfying the Kronecker delta property, either fully or weakly. Zambrano-Carrillo et al. [19] employed Moving Kriging (MK) functions, Ullah et al. [165,166] used maximum-entropy (max-ent) functions with weak Kronecker delta behaviour at the boundaries, and Zhang et al. [135] introduced MLS approximations with linearly varying support sizes to recover the property at the interface. Beyond hybrid coupling, the lack of the Kronecker delta property in standard EFG formulations has given rise to alternative strategies conceived to allow the imposition of essential (Dirichlet-type) boundary conditions when direct prescription of nodal values is not possible. In this context, penalty-based methods [12–16], Lagrange multipliers [12,72,73,78,79], and Nitsche’s [11,55,144] method have been employed, with the latter providing a more consistent and accurate enforcement without requiring excessively large penalty parameters. Alternatively, the use of shape functions based on such as Moving Kriging (MK) [145,146,160], interpolating MLS [60,83,86], or Local Maximum Entropy (LME) [43,136,155,165,166] formulations enables the straightforward imposition of prescribed nodal values, while retaining the intrinsic advantages of EFG in terms of high-order continuity and flexible node placement.
The present review aims to provide a detailed yet conceptually accessible discussion of the main theoretical and computational aspects underpinning the development and practical implementation of EFG formulations. In particular, attention is given to the construction of shape functions—including MLS, IMLS, MK, and LME approximations—their numerical integration and computational efficiency, and the strategies to impose essential boundary conditions, including penalty, Lagrange multiplier, and Nitsche-based approaches. Moreover, the review addresses the extension of EFG methods from classical linear elasticity problems to more complex scenarios, such as non-linear heat transfer with phase change, coupled fluid–heat transport, and multiphysics applications, highlighting the tailored formulations designed for such purposes. Finally, recent Chimera-Type hybrid FEM–EFG approaches are discussed [19,42], emphasising their ability to achieve seamless coupling and accurate transfer of field variables between overlapping domains without requiring a prescribed topological relationship.
Shape Functions in EFG Methods
Element-Free Galerkin (EFG) formulations are developed upon the weak form of the governing equations, which necessitates the construction of shape functions for the spatial approximation of the field variables. Unlike FEM, the construction of shape functions in EFG methods is not subjected to a geometric parametrisation of the problem domain at the local or elemental level. There is no need for a prescribed connectivity, and the evaluation of physical variables is performed within a moving local support domain by continuously capturing and weighting the information of neighbouring nodes involved in the construction of the shape functions at each point of interest. This provides greater flexibility in node placement and facilitates the treatment of problems involving large deformations, evolving boundaries, or discontinuities. Nevertheless, it has also introduced several numerical challenges concerning the imposition of essential boundary conditions and the computational cost associated with neighbour searching and matrix assembly.
In this section, several advanced techniques for constructing shape functions in EFG methods are reviewed. The discussion begins with the MLS approach, which constitutes the original basis of the EFG method, and proceeds with its subsequent enhancements, including the improved MLS, interpolating MLS, and complex-variable-based MLS variants. Following these, the Reproducing Kernel Particle Method (RKPM) is examined. Although originally conceived as a standalone framework, it has been recently recognised as a procedure for shape functions construction within EFG frameworks. Further developments, such as MK and LME approximations, are also examined, with particular emphasis on their fulfilment of the Kronecker delta property, which is fully satisfied in MK formulations and only weakly enforced at the boundaries in max-ent schemes.
Moving Least Squares (MLS) Approximations
The MLS approximations constitute the foundational technique for the construction of shape functions in EFG formulations, and still remain one of the most widely employed schemes for that purpose. Its fundamental idea is to approximate each scalar field variable u(x→) at any point x→ of a given computational domain Ω as a locally weighted least-squares fit of nodal parameters within a moving local support domain.
The MLS approximation of a scalar field is expressed as [30,90]
uh(x→)=qT(x→)a(x→)∀x→∈Ωwhere q(x→)=[1,x,y,z,…]T is a complete basis of polynomial terms (typically up to first or second order), and a(x→) is a vector of coefficients that depend on the local coordinate x→. These coefficients are determined by minimising a weighted discrete least-squares functional [30,90]
J(a(x→))=∑I=1nW^(x→−x→(I))[qT(x→(I))a(x→)−u(I)]2,where n is the number of nodes I supporting the approximation at x→. W^(x→−x→(I)) is a non-negative weight function of the node I–evaluated at x→–, whereas uI is the corresponding nodal parameter.
Minimising J with respect to a(x→) yields the following system of equations:A(x→)a(x→)=B(x→)u,with
A(x→)=∑I=1nW^(x→−x→(I))q(x→(I))qT(x→(I)),andB(x→)=[W^(x→−x→(1))q(x→(1)),…,W^(x→−x→(n))q(x→(n))].
Assuming A(x→) is invertible, the coefficient vector is obtained as
a(x→)=A−1(x→)B(x→)u,where uT=[u(1),u(2),…,u(n)]. By substituting (5) in (1), the field variable can be expressed in terms of the nodal parameters as
uh(x→)=∑I=1nφ(I)(x→)u(I),where
φ(I)(x→)=qT(x→)A−1(x→)W^(x→−x→(I))q(x→(I))are the MLS shape functions. Although these approximations do not satisfy the Kronecker delta property, they possess high smoothness (typically C1 or higher) and exhibit excellent approximation capabilities with properly chosen weight functions W^(x→−x→(I)) and support sizes. It is worth noting that the shape and dimensions of the support domains may vary depending on the nodal distribution, and also the geometric and dimensional characteristics of the problem. Circular or spherical supports are often adopted for regular [19,41] or moderately irregular [11,42] nodal layouts, whereas rectangular [41,42,107,113], cylindrical [10,11,41,42], hexahedral [9,19,90], or even polygonal and polyhedral [156,157,167] supports are preferred when the nodal spacing exhibits marked directional variations. In such cases, the support size can be independently adjusted along each coordinate direction to capture the local anisotropy of the nodal arrangement. Accordingly, both the weight function W^(x→−x→(I)) and the corresponding normalised distance measure are defined according to the chosen support geometry. The support size is typically defined as a multiple of the local nodal spacing to ensure sufficient overlap between neighbouring supports, which is essential to guarantee numerical stability and accuracy.
It is important to distinguish between the support domain and the influence domain, two notions that are often used interchangeably in the literature but have distinct meanings depending on the adopted convention [12]. The support domainΩs(x→) is defined for each point of evaluationx→ as the region encompassing all the nodes whose weight functions are nonzero at that location, i.e.,
Ωs(x→)={x→(I)∈Rd|W^(x→−x→(I))≠0}.
Conversely, the influence domainΩi(I) is associated with each node I and represents the spatial region over which its weight function contributes to the approximation,
Ωi(I)={x→∈Rd|W^(x→−x→(I))≠0}.
By definition, a node x→(I) influences a point x→ if and only if x→(I)∈Ωs(x→), which is equivalent to stating that x→∈Ωi(I). This dual relationship underpins the local nature of the Moving Least Squares (MLS) approximation, ensuring that only nearby nodes contribute to the construction of shape functions at a given point. This differentiation is not merely conceptual since it has a crucial role in defining the local nodal connectivity, and consequently in the assembly of the moment matrix involved in the MLS interpolation. The concepts of influence and support domain are depicted in Fig. 1.
Distinction between influence and support domains. Each node <inline-formula id="ieqn-25"><mml:math id="mml-ieqn-25"><mml:msup><mml:mrow><mml:mover><mml:mi>x</mml:mi><mml:mo stretchy="false">→</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula>, <inline-formula id="ieqn-26"><mml:math id="mml-ieqn-26"><mml:msup><mml:mrow><mml:mover><mml:mi>x</mml:mi><mml:mo stretchy="false">→</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula>, and <inline-formula id="ieqn-27"><mml:math id="mml-ieqn-27"><mml:msup><mml:mrow><mml:mover><mml:mi>x</mml:mi><mml:mo stretchy="false">→</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mn>3</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula> has an associated influence domain within which it can exert a contribution. Nevertheless, only nodes <inline-formula id="ieqn-28"><mml:math id="mml-ieqn-28"><mml:msup><mml:mrow><mml:mover><mml:mi>x</mml:mi><mml:mo stretchy="false">→</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula>, and <inline-formula id="ieqn-29"><mml:math id="mml-ieqn-29"><mml:msup><mml:mrow><mml:mover><mml:mi>x</mml:mi><mml:mo stretchy="false">→</mml:mo></mml:mover></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mn>3</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula> are used in to define the support domain and to construct shape functions for approximation at the interest point <inline-formula id="ieqn-30"><mml:math id="mml-ieqn-30"><mml:mrow><mml:mover><mml:mi>x</mml:mi><mml:mo stretchy="false">→</mml:mo></mml:mover></mml:mrow></mml:math></inline-formula>. It is worth mentioning that <inline-formula id="ieqn-31"><mml:math id="mml-ieqn-31"><mml:mrow><mml:mover><mml:mi>x</mml:mi><mml:mo stretchy="false">→</mml:mo></mml:mover></mml:mrow></mml:math></inline-formula> is not necessarily a node position, but any location within the problem domain.
The weight function W^(x→−x→(I)) is crucial in the definition of the local approximation, as it determines the extent and smoothness of the nodal contribution within its influence domain. Each node possesses an associated nodal basis function that is nonzero only inside its influence domain Ωi(I), where W^(x→−x→(I)) attains its maximum at the nodal position x→(I) and decreases monotonically and smoothly to zero at the influence domain boundary. Consequently, the weight function controls both the degree of locality and the smoothness of the resulting approximation. Conversely, the shape functionsφ(I)(x→) are constructed as combinations of these nodal basis functions through the MLS formulation, defining the actual approximation of the field variable at each evaluation point.
Commonly used weight functions include Gaussian distribution-based, quadratic, cubic, and quartic splines, and the compactly supported quartic weight function, which is very similar to the cubic splines and possesses second-order reproducing capacity [12]. The specific choice of weight function and its parameters has a strong influence on the approximation accuracy, the conditioning of the moment matrix, and the overall stability of the numerical formulation.
Improved MLS Approximations
The mainstream concept of IMLS approximations was first introduced by Lu et al. [31], who proposed the use of locally weighted orthogonal basis polynomials in order to simplify the evaluation of the moment matrix A(x→) in Eq. (4). The construction of an orthogonal vector of basis polynomials p(x→)=[p1(x→),p2(x→),…,pn(x→)]T within each local support domain gives rise to a diagonal moment matrix, which eliminates the need for its numerical inversion at every evaluation point. This improvement considerably enhances computational efficiency while preserving the smoothness and approximation accuracy characteristic of the standard MLS approaches. Building upon this idea, Kaljevié and Saigal [32] subsequently adopted the concept of Lu et al. and referred to it as IMLS approximations. The incorporation of this approach into the EFG framework was therefore termed the Improved EFG (IEFG) method, formalising the nomenclature for further works and demonstrating the practical benefits of this procedure in terms of computational efficiency. In this sense, further studies demonstrated that this procedure to construct shape functions is 30% faster compared to the standard MLS approximations [15,41,63].
The orthogonal polynomial basis vector is constructed from the components of a standard polynomial basis vector q(x→) as
p1=1,pi=qi−∑j=1i−1(qi,pj)(pj,pj),where the inner product between any pair of polynomial terms f(x→) and g(x→) is given by
(f,g)=∑I=1nW^(x→−x→(I))f(x→(I))g(x→(I)).
The orthogonal polynomial basis vector constructed using (10) yields the diagonal moment matrix with components [11,15,168]
Aij(x→)=(pi,pj)=∑I=1nW^(x→−x→(I))pi(x→(I))pj(x→(I))={0,i≠j,(pi,pi),i=j,(i,j=1,2,3,…,m).
Since the direct achievement of a diagonal moment matrix allows to circumvent the need for computing its inverse, the shape function for each node I can be merely computed as
φ(I)(x→)=W^(x→−x→(I))∑i=1mpi(x→)pj(x→(I))(pi,pi),where m is the number of polynomial terms. Please note that the definitions of n (number of nodes supporting the approximation at a position x→) and m (number of polynomial terms) will remain during the entire communication.
Interpolating MLS Approximations
In principle, the interpolating MLS approximations can be derived from the standard MLS by using singular weighting functions [83,86,169]. The development of interpolating MLS emerged in order to overcome the lack of Kronecker delta property in the standard MLS approximations, but the singularity of the weighting functions at the corresponding node positions of the yields to ill-conditioned moment matrices. A detailed discussion concerning this inherent instability of the interpolating MLS can be found in the work of Li and Wang [169], where a stabilised approach based on shifted and scaled polynomial basis is also proposed and analysed. Improved interpolating MLS (IIMLS) approximations, able to suppress the requirement of using singular weight functions, were proposed by Wang et al. [170], and this will be the procedure to be presented in this section.
The IIMLS starts from a standard vector of polynomial basis q(x→) with q1(x→)=1, from which a new set of polynomial terms is constructed at the support of a given point x→ as
pi(x→∗)=qi(x→∗)−∑I=1nv(x→,x→(I))qi(x→(I)),with:v(x→,x→(I))=χ(x→,x→(I))∑J=1nχ(x→,x→(J))andχ(x→,x→(I))=∏J≠I‖x→−x→(J)‖2∏J≠I‖x→(I)−x→(J)‖2,where x→∗ is a point in the local support of x→. The same transformation given in (14) is now applied to the scalar field variable u(x→), i.e.,
u~(x→∗)=u(x→∗)−∑I=1nv(x→,x→(I))u(x→(I)).
It is worth noting that v(x→(I),x→(J))=δIJ and ∑I=1nv(x→,x→(I))=1. The next step consists in performing the MLS approximation of u~(x→), which is
u~L(x→,x→∗)=∑i=2mpi(x→∗)a~i(x→)⏟pT(x→∗)a^(x→).
Please note that the summation (17) start at i=2, since it can be demonstrated from (14) and the partition unity property of v(x→,x→(I)) that p1(x→∗)=0. The coefficients a~i(x→) are obtained by minimisation of the functional:J^=∑I=1nW^(x→−x→(I))[pT(x→(I))a~(x→)−u~(x→(I))]2=∑I=1nW^(x→−x→(I))[pT(x→(I))a~(x→)+∑J=1nv(x→,x→(J))u(x→(J))−u(x→(I))]2.
The minimisation of J~ with respect to a~(x→) yields
dJ^da^=0=∑I=1nW^(x→−x→(I))p(x→(I))pT(x→(I))a~(x→)+∑I=1nW^(x→−x→(I))p(x→(I))[∑J=1nv(x→,x→(J))u(x→(J))−u(x→(I))],which can be rewritten as
A(x→)a~(x→)=B(x→)[I−V(x→)]u∴a~(x→)=A−1(x→)B(x→)[I−V(x→)]uwhere:A(x→)=∑I=1nW^(x→−x→(I))p(x→(I))pT(x→(I)),B(x→)=[W^(x→−x→(1))p(x→(1)),…,W^(x→−x→(n))p(x→(n))],whereas V(x→) is a n×n matrix made of n repetitions of the row
v(x→)=[v(x→,x→(1)),v(x→,x→(2)),…,v(x→,x→(n))],and I is the n×n identity matrix. Accordingly, the local approximation u~L(x→,x→∗) is
u~L(x→,x→∗)=pT(x→∗)A−1(x→)B(x→)(I−V(x→))u.
Finally, the substitution of (23) in the transformation (16) yields
uh(x→)=[pT(x→)A−1(x→)B(x→)(I−V(x→))+v(x→)]⏟φ(x→)u.
The achieved approximation fulfils the interpolating property and does not involve any singular weight function, so any standard weight function used in MLS can also be used for the IIMLS [149,170]. Although Wang et al. [170] claimed that the interpolating property of Eq. (24) is obvious, it is also fair to state that it is not strictly trivial. In fact, the operator (I−V(x→)) generates a vector of differences between the nodal field values and the local nodal value at x→=x→(I), i.e., (I−V(x→(I)))u=u−u(I)I. Since the MLS approximation exactly reproduces constant fields, any constant component of u (such as uII) is orthogonal to the polynomial space spanned by p(x→). Consequently, the term involving A−1(x→)B(x→)(I−V(x→)) vanishes at node x→(I), and only v(x→(I))u=u(I) remains, thus ensuring the interpolating property.
Reproducing Kernel Particle Method
Although the EFG method is traditionally built upon the MLS approximation, the Reproducing Kernel Particle Method (RKPM) has been recently recognised in the literature as a formally equivalent engine for constructing the underlying shape functions. As seen in the work of Cheng and Liew [171], RKPM is commonly treated as an independent meshfree method that utilizes a variational form to solve physical problems. On the other hand, Dehghan and Abbaszadeh [172] have formally categorised this approach as an RKPM-based EFG method. This shift acknowledges that when a global Galerkin weak form is used, the choice between MLS and RKPM becomes a matter of preference to construct the shape functions rather than a change in the fundamental numerical method.
The construction of RKPM shape functions starts from the definition of a continuous kernel approximation uh(x→) of a function u(x→) as follows [171,172]:uh(x→)=∫ΩqT(x→−x→′)b(x→)W^(x→−x→′)u(x→′)dx→′where W^(x→−x→′) is the kernel function with a compact support, and qT(x→−x→′)b(x→) is the correction function expressed as a linear combination of the basis qT(x→−x→′). The vector of unknown coefficients b(x→) is determined by enforcing the reproducing conditions after applying a trapezoidal integration rule of the continuous (25) kernel approximation over the nodes x→(I), which yields
b(x→)=M−1(x→)Hwhere the moment matrix M(x→) is
M(x→)=∑I=1nq(x→−x→(I))qT(x→−x→(I))W^(x→−x→(I))ΔV(I)and H=[1,0,0,…,0]T. Substituting the unknown coefficients in the trapezoidal integration of the continuous kernel approximation yields
uh(x→)=∑I=1nqT(x→−x→(I))M−1(x→)HW^(x→−x→(I))ΔV(I)⏟φ(I)(x→)u(I).
In this formulation, ΔV(I) represents the volume often associated with the Voronoi cell of each node I. It is important to note that for uniform nodal distributions, ΔV(I) becomes constant and these shape functions become numerically identical to the standard MLS.
Moving Kriging Interpolation
The implementation of the MK interpolation within the EFG framework was first introduced in the pioneering work of Gu [173], allowing the construction of shape functions fulfilling the Kronecker delta property. The MK interpolation is based on a combination of a linear regression model and stochastic departures [145,160,173,174]:uh(x→)=qT(x→)a+Z(x→),where q(x→) is the vector of polynomial basis functions, a is a vector of unknown coefficients, and Z(x→) is assumed to be a stochastic zero-mean process of variance σ2 and non-zero covariance given by
cov{Z(x→(I),x→(J))}=σ2R[R(x→(I),x→(J))],where R is the square n×n correlation matrix composed of the correlation functions R(x→(I),x→(J)) linking each pair (I,J) of the n nodes supporting the MK interpolation at x→. The most common and simplest correlation function is the Gaussian form:R(x→(I),x→(J))=e−θ‖x→(I)−x→(J)‖2.
The MK interpolation is obtained by minimising the following best linear unbiased predictor (BLUP) functional:J[uh(x→)]=E[uh(x→)−u(x→)]2,subjected to the constraint
E[uh(x→)]=E[u(x→)],where E[(⋅)]=∫Ω(⋅)dΩ/∫ΩdΩ. The minimisation process, after substituting Eq. (29) into Eqs. (32) and (33), yields
uh(x→)=[qT(x→)A+rT(x→)B]⏟φ(x→)u,with
A=(QTR−1Q)−1QTR−1,andB=R−1(I−QA),where Q is an n×m matrix defined as
Q=[q(x→(1)),q(x→(2)),…,q(x→(n))]T,whereas the vector r(x→) has components r(I)(x→)=R(x→(I),x→), so that r(x→(J))=R. The matrix A has dimensions m×n, as can be verified by inspection of Eq. (35). The vector of shape functions φ(x→) has components
φ(I)(x→)=∑k=1mqk(x→)AkI+∑k=1nrk(x→)BkI,from which it is straightforward to prove the Kronecker delta property of φ(I)(x→), i.e., φ(I)(x→(J))=δIJ. Indeed, the value of φ(I)(x→(J)) is given by
φ(I)(x→(J))=∑k=1mqk(x→(J))AkI+∑k=1nrk(x→(J))BkI,which, by virtue of qT(x→(J))=Q and r(x→(J))=R, can be expressed in matrix form as
φ(x→(J))=QA+RB.
Substituting B from Eq. (35) into Eq. (39) finally yields
φ(x→(J))=QA+RR−1(I−QA)=I,or equivalently,φ(I)(x→(J))=δIJ,thus confirming the Kronecker delta property. It should be noted that the construction of the MK shape functions is computationally more demanding than that of the standard MLS approximation. This is mainly due to the matrix inversions involved in the evaluation of both A and B in Eq. (35). In particular, the computation of A requires the inversion of the m×m matrix QTR−1Q, whereas B involves an additional n×n inversion through R−1. Consequently, the MK interpolation fulfils the Kronecker delta property, albeit at a higher computational cost. Additionally, Tu et al. [175] demonstrated that the matrices involved in the MK formulation become increasingly ill-conditioned as the nodal distribution is refined. To address this issue, they proposed a stabilised MK interpolation based on shifted and scaled polynomial basis functions:q^T(x→)=qT(x→−x→(e)Chh),where x→(e) denotes the centroid of the n local nodes used to construct the MK interpolation at x→, i.e.,
x→(e)=1n∑I=1nx→(I).
The parameters Ch and h correspond to a scaling constant and the minimum distance between any pair of the supporting nodes x→(I). The procedure to perform the MK interpolation remains the same, but with all operators defined in terms of the shifted and scaled polynomial basis. After performing the mapping q(x→)→q^(x→), Q→Q^, and R→R^, the components of Q^ and R^ are given by:Q^=[q^(x→(1)),q^(x→(2)),…,q^(x→(n))]T,and[R^](IJ)=R^(x→(I),x→(J))=e−θ(Chh)2‖x→(I)−x→(J)‖2.
According to Tu et al. [175], the optimal value of Ch for achieving the best conditioning of the matrices involved in the MK interpolation is 4.5. The correlation parameter θ has a strong influence on the accuracy of the MK approximation, whereby Zheng and Dai [174] proposed the expression θ=ω/lavg2 for this parameter. The measure lavg represents the average distance between the nodes supporting the approximation at x→, and good accuracy is commonly achieved with ω=0.1 [174–176].
Local Maximum Entropy Approximations
The construction of shape functions based on the principle of maximum entropy was first introduced by Sukumar [177] to address the underdetermined system of equations that arises in the construction of approximations over polygons with more than three sides in 2D problems, subjected to constant and linear reproducing constraints. Subsequently, Arroyo and Ortiz [178] proposed the Local Maximum Entropy (LME) formulation by introducing weight functions (or priors) into Sukumar’s original approach [177]. This modification enabled the construction of shape functions within a moving local support domain, Ωs(x→), defined by the neighbouring nodes with non-zero weight functions at the point of interest x→. Such approximations were then employed in the meshless solution of linear and non-linear elasticity problems. The weak Kronecker delta property of LME-based shape functions was also demonstrated, along with their correspondence with standard MLS approximations. Although Arroyo and Ortiz [178] used Gaussian weight functions in their pioneering work, the formulation was later generalised by Sukumar and Wright [179] to accommodate arbitrary weight functions.
The construction of LME approximations arises from the principle of determining a set of shape functions φ(I)(x→) for the nodes x→(I)∈Ωs(x→) such that
∑I=1nφ(I)(x→)=1and∑I=1nφ(I)(x→)x→(I)=x→,thereby fulfilling the constant and linear reproducing conditions [155].
For polygons with three sides (triangles) in 2D or polyhedra with four faces (tetrahedra) in 3D, Eq. (44) admits a unique solution. For cases involving a larger number of nodes defining the moving local support domain—as is typically the case in EFG methods—Eq. (44) becomes underdetermined, i.e., there are more unknowns φ(I)(x→) than available constraints. In such cases, the principle of maximum entropy is used to determine the unknown shape functions φ(I)(x→) by solving the following optimisation problem [165,180–182]:maxφ[H(φ(x→))]=−∑I=1nφ(I)(x→)ln(φ(I)(x→)W^(x→−x→(I))),subject to the constant and linear reproducing constraints in Eq. (44).
The rationale behind the LME formulation (45) is to ensure that the resulting shape functions φ(I)(x→) fulfil the reproducing conditions without introducing any additional, unwarranted assumptions about their values. In other words, the maximum-entropy solution represents the smoothest and most objective approximation consistent with the available information [178]. The solution of (44) subject to (45) yields [165,180–182]
φ(I)(x→)=W^(x→−x→(I))e−λ→⋅x→∗(I)∑J=1nW^(x→−x→(J))e−λ→⋅x→∗(J),where x→∗(I)=x→(I)−x→ are the shifted coordinates of the supporting nodes x→(I)∈Ωs(x→), and λ→ is the vector of Lagrange multipliers enforcing the linear reproducing constraints. This vector is λ→=[λx,λy] and λ→=[λx,λy,λz] in 2D and 3D problems, respectively. The vector of Lagrange multipliers is obtained from the dual of the optimisation problem (44) and (45), leading to the following non-linear system of equations [180,181]:f(λ→)=−∑I=1nW^(x→−x→(I))e−λ→⋅x→∗(I)x→(I)∑J=1nW^(x→−x→(J))e−λ→⋅x→∗(J)=x→.
This system is solved using the Newton–Raphson method, with the tangent operator (i.e., the Hessian matrix of the optimisation problem) given by [180,181]
H=∑I=1nφ(I)(x→)x→∗(I)⊗x→∗(I).
A more detailed and practical explanation of the Newton–Raphson procedure for solving Eq. (47) can be found in the PhD thesis of Ullah [183].
Solving Numerical Difficulties in EFG Methods
Although Element-Free Galerkin (EFG) methods exhibit several noteworthy numerical advantages—such as [12,19,42] (i) the possibility of more easily achieving higher-order approximations with continuous derivatives, (ii) greater flexibility in adding or removing nodes, (iii) the absence of any post-processing requirement to obtain smooth physical fields derived from the primary variables, and (iv) improved accuracy compared with piecewise polynomial approximations used in standard FEM formulations—these methods also pose several implementation challenges.
EFG formulations can experience issues common to mesh-based techniques, such as volumetric locking in nearly incompressible media [34,106,184,185], shear locking in bending-dominated problems [31,35], and instabilities in advection-dominated transport phenomena scenarios [9,186–188]. EFG methods can also introduce additional difficulties specific to their formulation such as a higher computational cost mainly to the following aspects: the repeated inversion of local moment matrices [12,189,190], the high-order numerical integration required for non-polynomial shape functions [44–46,49], and the searching for neighbouring nodes at each integration point [39–41,43]. Other well-known issues include the need for special techniques to impose essential boundary conditions when using shape functions that do not satisfy the Kronecker delta property [11,12], the sensitivity of the results to the definition of the support domains—particularly in irregular nodal distributions [11,12,15]—and the treatment of material discontinuities [33,90,191,192]. This section discusses several strategies that have been developed to effectively address these difficulties.
Imposition of Essential Boundary Conditions and Treatment of Material Discontinuities
In EFG formulations based on shape functions that fulfil the Kronecker delta property (φ(I)(x→(J))=δIJ), the accomplishment of essential (Dirichlet) boundary conditions is straightforward, as they can be directly imposed in the form of prescribed nodal values. However, shape functions such as the MLS approximations do not possess this property. Consequently, the field variable at a node does not coincide with its nodal parameter, and essential boundary conditions cannot be imposed by merely prescribing nodal values as in mesh-based methods. To address this limitation, several specialised techniques have been developed for the imposition of Dirichlet boundary conditions. The most widely used approaches are the technique of Lagrange multiplier, the penalty method, and Nitsche’s formulation.
The method of Lagrange multipliers provides an exact enforcement of essential boundary conditions but enlarges the system of equations and may deteriorate its conditioning and sparsity. On the other hand, the penalty method preserves the system size and symmetry while maintaining a positive definite stiffness matrix. Nevertheless, its effectiveness depends on the appropriate choice of the penalty parameter. Excessively large values can lead to ill-conditioning, whereas smaller ones may result in a poor fulfilment of the essential boundary conditions. Finally, Nitsche’s method offers a more consistent and theoretically rigorous alternative by weakly enforcing the Dirichlet conditions through a symmetric modification of the weak form, providing improved accuracy and stability without introducing additional unknowns.
For the sake of generality, the discussion of these methods is presented with reference to a generic boundary value problem expressed in weak form as
a(δu,u)=ℓ(δu)∀δu∈𝒱0,where a(δu,u) denotes the bilinear form associated with the governing differential operator, and ℓ(δu) represents the corresponding linear functional.
Lagrange Multiplier Method
In this procedure, the weak form (49) is augmented, introducing a Lagrange multiplier field λ(x→) defined along the Dirichlet boundary Γu. The modified weak form reads [12,37,73,193]
a(δu,u)+∫Γuδλ(u−u¯)dΓ+∫ΓuλδudΓ=ℓ(δu),where λ enforces the constraint u=u¯ in a variationally consistent manner.
The Lagrange multiplier λ(x→) acts as an additional unknown field, which must be interpolated over the essential boundary. For a curvilinear coordinate s defined on Γu, the Lagrange multiplier field can be expressed as
λh(s)=∑J=1nsN(J)(s)λ(J),s∈Γu,where ns denotes the number of boundary nodes used for interpolation, and N(J)(s) are standard Lagrange interpolation functions.
Substituting the discrete forms uh of (6) and λh of (51) into the augmented weak form (50) leads to the following matrix system:[KΛTΛ0][uλ]=[fg],where Λ represents the coupling between boundary and domain degrees of freedom.
This formulation ensures complete fulfilment of essential boundary conditions on Γu. However, the additional unknowns yield a larger system of equations. The matrix is no longer positive definite, whereby conditioning issues are introduced. These features have motivated the development of alternative, more efficient enforcement strategies, such as the penalty and Nitsche methods.
Penalty Method
The penalty method provides an alternative and simpler approach for the enforcement of essential boundary conditions in EFG formulations. In this technique, the constraints are imposed approximately by introducing a term that penalises any deviation of the numerical solution from the prescribed boundary values. Unlike the Lagrange multipliers technique, the penalty method does not introduce additional field variables that enlarge the algebraic system. This method instead modifies the stiffness matrix directly, maintaining its size and positive definiteness.
In the penalty approach, the weak form is augmented with a term that penalises the deviation between the approximate solution uh and the prescribed value u¯ along Γu as follows [12,13,16,106]:a(δu,u)+∫Γuαδu(u−u¯)dΓ=ℓ(δu),where α>0 is the penalty parameter that controls the strength of the constraint enforcement. The higher the value of α, the closer u approaches u¯ on Γu. However, excessively large values may lead to ill-conditioning of the global system of equations.
The corresponding discrete system of equations can be expressed as
(K+Kα)u=f+fα,where K and f denote the global stiffness matrix and external force vector obtained from the domain integrals, whereas Kα and fα are the penalty contributions for the approximate imposition of the Dirichlet-boundary conditions.
The main advantages of the penalty method are its simplicity and computational efficiency, as the system matrix remains symmetric, positive definite, and of unchanged dimension. However, the method only enforces the essential conditions approximately. The accuracy and stability of the method is highly sensitive to an appropriate choice of the penalty parameter α since: excessively small values lead to poor constraint satisfaction, whereas very large ones can deteriorate the conditioning of the system. Nevertheless, an appropriate choice of the penalty parameter provides a suitable trade-off between numerical stability and the accuracy of boundary conditions enforcement.
Nitsche’s Method
Nitsche’s method can be interpreted as a consistent extension of the penalty formulation that allows the weak enforcement of Dirichlet boundary conditions, without either introducing additional unknowns or suffering from the conditioning issues associated with large penalty parameters. This approach has been successfully applied to both mesh-based and meshfree formulations due to its consistency, stability, and flexibility.
The method augments the weak form by adding terms that weakly enforce the Dirichlet condition, while preserving both the symmetry and consistency of the formulation. For a generic boundary value problem, the modified weak form reads [55,123,144,158]
a(δu,u)−∫Γuδu(κ∇u⋅n^)dΓ+∫Γu(κ∇δu⋅n^)(u−u¯)dΓ+∫Γuαδu(u−u¯)dΓ=ℓ(δu),where n^ denotes the unit outward normal vector to Γu, κ is the transport coefficient included in the diffusive term of a(δu,u), and α is a stabilisation parameter that performs a role similar to the penalty factor in Eq. (53).
The first two additional boundary terms in Eq. (55) ensure consistency of the weak formulation, whereas the third (symmetric) term provides stability. Nitsche’s formulation enforces the boundary conditions exactly in the limit of mesh refinement, without the need for excessively large stabilisation parameters.
The discrete form of Eq. (55) can be written as
(K+KN)u=f+fN,where KN and fN represent the symmetric Nitsche contributions associated with the boundary integrals. These contributions depend on both the shape functions and their normal derivatives evaluated along Γu.
The main advantages of Nitsche’s method are its consistency, improved accuracy, and the possibility of achieving optimal convergence rates without introducing additional variables. The system matrix remains both symmetric and well-conditioned, whereby this approach is often preferred over the penalty method for the weak imposition of Dirichlet boundary conditions in meshfree formulations. The stabilisation parameter α can be selected following analytical or numerical criteria to guarantee coercivity, typically as a function of the local stiffness and characteristic nodal spacing along the boundary.
In addition to the aforementioned variational methods, other strategies have been developed to address the lack of the Kronecker delta property in EFG approximations. The full transformation method [194,195] introduces a matrix that maps the meshfree degrees of freedom to the actual physical nodal values. This allows the essential boundary conditions to be imposed directly, as in conventional FEM, circumventing the need for additional interface integrals or multipliers. Alternatively, Kaljević and Saigal [32] proposed the use of singular weight functions to achieve interpolating properties with MLS approximations. Using kernels that tend to infinity at the nodal locations yields interpolating shape functions. Although this property can be enforced throughout the entire domain, its application can be specifically targeted at the boundary nodes to facilitate a straightforward enforcement of essential conditions without auxiliary variables. More recently, a variationally consistent framework has been proposed based on the reproducing kernel approach in the context of the Hellinger-Reissner variational principle [196]. This is a mixed formulation specifically developed in the framework of linear elasticity problems, where the displacement and stress fields are approximated independently to allow the essential boundary conditions to be naturally incorporated into the weak form through the stress-displacement coupling. This approach shares a similar structure with Nitsche’s method but offers the significant advantage of completely eliminating the need for the problem-dependent artificial parameters (α) required for stability in Eq. (55). Such parameter-free enforcement ensures stability and optimal convergence, representing a robust alternative for nearly incompressible elastic media and other elliptic systems where traditional stabilisation approaches may be challenging.
Application to Material Discontinuities
The aforementioned weak enforcement strategies can also be extended to impose interface conditions across material discontinuities. In such cases, the physical domain Ω is divided into two subdomains Ω+ and Ω−, separated by an internal interface Γi over which continuity of the field variable u and the corresponding diffusive flux −κ∇u⋅n^ must be guaranteed. The continuity conditions across Γi can be written as
⟦u⟧=u+−u−=0,⟦κ∇u⋅n^⟧=0,where ⟦⋅⟧ denotes the jump operator and n^ is the unit normal to Γi oriented from Ω− to Ω+.
The enforcement of the interface conditions in Eq. (57) follows the same principles used for Dirichlet boundaries:
Lagrange multipliers: The weak form is augmented with interface terms involving a multiplier field λ defined on Γi, enforcing ⟦u⟧=0 in a variationally consistent way [33,191]:a(δu,u)+∫Γiδλ⟦u⟧dΓ+∫Γiλ⟦δu⟧dΓ=ℓ(δu).This approach ensures exact satisfaction of continuity but introduces additional unknowns associated with λ.
Penalty method: The interface continuity is enforced approximately by penalising the jump in u across Γi [12]:a(δu,u)+∫Γiα⟦δu⟧⟦u⟧dΓ=ℓ(δu),where α is a penalty parameter controlling the strength of the constraint.
Nitsche’s method: A consistent and stable alternative is obtained by symmetrically adding interface terms analogous to those introduced for Dirichlet boundaries [197]:a(δu,u)−∫Γi⟦δu⟧{{κ∇u}}⋅n^dΓ+∫Γi{{κ∇δu}}⋅n^⟦u⟧dΓ+∫Γiα⟦δu⟧⟦u⟧dΓ=ℓ(δu),
where {{⋅}} denotes the average operator across Γi. The first two interface terms ensure consistency, whereas the last term guarantees stability.
These formulations enable a unified and flexible treatment of discontinuous material domains, allowing jumps in material parameters to be handled naturally within the weak formulation. The choice among Lagrange multipliers, penalty, or Nitsche’s approach depends on the desired trade-off between accuracy, computational cost, and conditioning of the resulting algebraic system.
It is important to remark that the bilinear and linear operators a(δu,u) and ℓ(δu) must be integrated considering the appropriate nodal support for each subdomain, as depicted in Fig. 2. The integrations over Ω− are performed using shape functions constructed only with the nodes belonging to Ω− and the interface nodes located on Γi, whereas those over Ω+ are evaluated using the nodes of Ω+ and the same shared interface nodes. This ensures that the support of each trial and test function remains confined to its corresponding subdomain. Integrals defined along the interface Γi involve quantities computed from both sides: the jump and average operators are evaluated using the nodal fields associated with Ω− (blue nodes) and Ω+ (red nodes), respectively. This distinction guarantees a consistent assembly of the weak form, preserving the correct coupling and continuity between the two material regions.
Representation of the computational domain divided into two subdomains, <inline-formula id="ieqn-225"><mml:math id="mml-ieqn-225"><mml:msup><mml:mi mathvariant="normal">Ω</mml:mi><mml:mo>−</mml:mo></mml:msup></mml:math></inline-formula> (blue) and <inline-formula id="ieqn-226"><mml:math id="mml-ieqn-226"><mml:msup><mml:mi mathvariant="normal">Ω</mml:mi><mml:mo>+</mml:mo></mml:msup></mml:math></inline-formula> (red), separated by the internal interface <inline-formula id="ieqn-227"><mml:math id="mml-ieqn-227"><mml:msub><mml:mi mathvariant="normal">Γ</mml:mi><mml:mi>i</mml:mi></mml:msub></mml:math></inline-formula> (black). The blue and red circles denote the nodes belonging exclusively to <inline-formula id="ieqn-228"><mml:math id="mml-ieqn-228"><mml:msup><mml:mi mathvariant="normal">Ω</mml:mi><mml:mo>−</mml:mo></mml:msup></mml:math></inline-formula> and <inline-formula id="ieqn-229"><mml:math id="mml-ieqn-229"><mml:msup><mml:mi mathvariant="normal">Ω</mml:mi><mml:mo>+</mml:mo></mml:msup></mml:math></inline-formula>, respectively, whereas black circles correspond to the nodes shared by both subdomains and located on <inline-formula id="ieqn-230"><mml:math id="mml-ieqn-230"><mml:msub><mml:mi mathvariant="normal">Γ</mml:mi><mml:mi>i</mml:mi></mml:msub></mml:math></inline-formula>. Integrals over <inline-formula id="ieqn-231"><mml:math id="mml-ieqn-231"><mml:msup><mml:mi mathvariant="normal">Ω</mml:mi><mml:mo>−</mml:mo></mml:msup></mml:math></inline-formula> and <inline-formula id="ieqn-232"><mml:math id="mml-ieqn-232"><mml:msup><mml:mi mathvariant="normal">Ω</mml:mi><mml:mo>+</mml:mo></mml:msup></mml:math></inline-formula> are evaluated using shape functions constructed from their respective nodal sets, including the interface nodes. Quantities involved in the interface integrals over <inline-formula id="ieqn-233"><mml:math id="mml-ieqn-233"><mml:msub><mml:mi mathvariant="normal">Γ</mml:mi><mml:mi>i</mml:mi></mml:msub></mml:math></inline-formula>, such as jumps and averages, are computed using the blue-side values for the minus domain and the red-side values for the plus domain.
It is worth mentioning that the interface integrals involved in the methods of Lagrange Multipliers, Penalty, and Nitsche are primarily required for EFG formulations based on shape functions that do not fulfil the Kronecker delta property. Alternatively, material interfaces can be accurately modelled by using shape functions that possess this property at the interface Γi. While standard MLS approximations lack this feature, alternative schemes such as Moving Kriging [173], LME [178], or Interpolating MLS [170] allow for the direct nodal enforcement of continuity. The requirement ⟦u⟧=0 is naturally fulfilled through the shared nodal degrees of freedom at the interface Γi, effectively connecting the subdomains Ω+ and Ω− without the need for additional interface integrals or stabilisation parameters. In this framework, the gradient or flux discontinuity across the interface—arising from different material properties—is captured naturally by the assembly of independent shape function supports on either side of Γi. This approach significantly simplifies the implementation for multi-material problems, as the interface conditions are implicitly fulfilled by the interpolating nature of the approximation space.
Efficient Numerical Integration Schemes
In EFG methods, the accurate evaluation of domain and boundary integrals remains one of the main numerical challenges. The non-polynomial character of the shape functions and the overlap of support domains make the exact integration of the weak form virtually impossible, whereby the choice of a suitable quadrature scheme is crucial for ensuring accuracy, convergence, and computational efficiency. EFG methods often demand high-order quadrature schemes, since low-order integration rules can lead to non-converging and unstable solutions. Recent theoretical research by Wu and Wang [198] has established that this phenomenon arises from losing the Galerkin orthogonality condition due to integration errors. In such a fundamental work, it has formally been demonstrated that the overall convergence of EFG methods is governed by a synergy between the interpolation error (associated with the basis functions) and the integration error (associated with the quadrature). Consequently, the development of efficient domain integration algorithms specifically tailored for EFG methods is not only motivated by computational cost but by the mathematical requirement of matching the integration consistency with the approximation order to restore optimal convergence. The following section summarises the main approaches proposed in this context, highlighting their underlying principles and comparative performance.
The first attempt to improve the computational efficiency of domain integration in EFG methods was proposed by Beissel and Belytschko [199] in the context of elliptic problems in linear elasticity, and is known as the direct nodal integration (DNI) method. In this procedure, the integration domain associated with each node is defined through a Voronoi tessellation of the problem domain. Each Voronoi cell represents the region of integration for the corresponding node, and the integrals are approximated by single-point evaluations at the nodal positions. The integration weights are determined by the area (or volume) of the Voronoi cell for interior nodes and by the length (or surface area) of the boundary portions for nodes located on the traction boundaries. This strategy considerably simplifies the numerical integration process and removes the dependence on any auxiliary background mesh, thereby significantly reducing the computational cost. However, although the method proved highly efficient, it suffered from stability issues and exhibited sub-optimal convergence behaviour. This has motivated researchers to propose methods devoted to eliminate such instabilities, which include: adding squared residuals-based stabilisation terms to yield a residual-based DNI (RDNI) scheme [199], introducing the strain field from a first-order Taylor of the displacement field approximation to obtain a naturally stabilised DNI (NDNI) approach [200], or the least squares stabilisation term used in the modified DNI (MDNI) [201]. Despite the aforementioned improvements, these methods still inherit some fundamental limitations from the original DNI formulation. In particular, none of them is able to fully satisfy the linear patch test. This indicates that the consistency of the approximated strain field is not entirely recovered, although accurate displacement solutions can be obtained. The stress predictions remain sub-optimal, especially near the boundaries. Moreover, the computation of higher-order derivatives required in these formulations is often time-consuming and susceptible to accumulated numerical errors. A comprehensive discussion in this regard can be found in the work of Luo et al. [47]. These drawbacks have motivated the development of more robust integration strategies capable of restoring both stability and consistency, such as the so-called stabilised conforming nodal integration (SCNI) methods.
The SCNI approach introduced in the seminal work of Chen et al. [202] constitutes the first variationally consistent integration framework for EFG methods. In this approach, the shape functions gradient ∇φ(I) is replaced by a smoothed counterpart ∇~φ(I) defined within integration subdomain Ωg(I) associated with each node. It is worth remarking that the integration domain Ωg(I) of a node must not be confused with its influence domain Ωi(I), i.e., that region where φ(I)(x→)≠0. The smoothed operator is defined as an averaged (or “smoothed”) gradient obtained as
∫Ωg(I)∇~φ(I)(x→)dΩ=∫Γg(I)φ(I)(x→)⋅n^dΓ,where φ(I) denotes the shape function associated with node I, and n^ is the outward unit normal to the boundary Γg(I) of the nodal integration subdomain. The equality of Eq. (61) is known as the integration constraint, and ensures that the numerical solution of the weak form is consistent with the Galerkin formulation. This means that linear fields are exactly reproduced within each nodal subdomain, which guarantees first-order consistency of the discretisation and thus fulfilment of the linear patch tests. By satisfying this constraint, the SCNI method eliminates the instability and suboptimal convergence behaviour of direct nodal integration. However, the smoothed strain evaluated at the centre of each nodal integration cell can only reproduce a constant strain field within. The subdomain and corresponding boundaries of each node x(I) are commonly defined according to the conforming Voronoi diagram of the nodes distribution, as depicted in Fig. 3. Consequently, SCNI can only provide accuracy and convergence comparable to linear finite elements even if a quadratic approximation is used. Duan et al. [203] introduced the Quadratically Consistent Integration (QCI) framework to overcome such a limitation, which has been achieved by extending the variationally consistent concept of SCNI via the enforcement of both the differentiation of approximation consistency (DAC) and the discrete divergence consistency (DDC). The DAC condition requires that the derivatives of the shape functions φ(I) preserve the consistency of the polynomial basis p(x→), expressed as
∇p(x→)=∑I=1np(x→(I))∇φ(I)(x→),which ensures the exact reproduction of the chosen basis under differentiation. In turn, the DDC imposes that the numerical derivatives ∇φ(I) satisfy the divergence theorem in a discrete sense, providing compatibility between φ(I) and their gradients:∫Ωg(I)∇φ(I)(x→)q(x→)dΩ=∫Γg(I)φ(I)(x→)q(x→)n^dΓ−∫Ωg(I)φ(I)(x→)∇q(x→)dΩ,where q(x→) is a smooth test function constructed from the derivatives of the polynomial basis p(x→).
Schematic representation of nodal integration subdomains defined through a conforming Voronoi tessellation of the node distribution. In the SCNI framework, each Voronoi cell serves as a smoothing domain <inline-formula id="ieqn-250"><mml:math id="mml-ieqn-250"><mml:msubsup><mml:mi mathvariant="normal">Ω</mml:mi><mml:mi>g</mml:mi><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>I</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup></mml:math></inline-formula> for the evaluation of the smoothed gradient <inline-formula id="ieqn-251"><mml:math id="mml-ieqn-251"><mml:mrow><mml:mover><mml:mi mathvariant="normal">∇</mml:mi><mml:mo stretchy="false">~</mml:mo></mml:mover></mml:mrow><mml:msup><mml:mi>φ</mml:mi><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>I</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup></mml:math></inline-formula>. Similar Voronoi-based partitions have been subsequently adopted in other integration schemes such as QCI, RKGSI, and CEFG, owing to their convenient and mesh-independent definition of nodal subdomains.
In its original form, the method employed three quadrature points per integration cell (QC3) to compute smoothed nodal derivatives that fulfil both the quadratic DAC and DDC. Subsequently, Duan et al. [204,205] generalised this concept to a one-point quadrature scheme (QC1) by preserving quadratic consistency via the incorporation of higher-order derivatives evaluated at the cell centre. This formulation maintains second-order consistency while significantly reducing computational cost, and can exactly pass both the linear and quadratic patch tests. Based on these developments, a more general integration framework was later proposed by Chen et al. [51] under the name Variationally Consistent Integration (VCI). This approach extends the concept of variational consistency beyond the quadratic level achieved by QCI, providing a unified variational basis for constructing integration schemes of arbitrary order. Instead of relying on specific smoothing or correction procedures, VCI enforces that the discrete weak form exactly reproduces the continuous variational equations for polynomial fields up to a prescribed degree. Accordingly, it guarantees the exact satisfaction of higher-order patch tests while maintaining computational efficiency and stability. The SCNI and QCI schemes can be regarded as particular cases of the general VCI framework for first- and second-order consistency, respectively. Although the VCI framework provides a robust, theoretically general formulation, its practical implementation is considerably more involved than in SCNI or QCI. The method requires solving local variational problems within each integration subdomain to determine the consistent projection operators, which increases the formulation complexity and computational cost. A more general variationally consistent approach is provided by the Consistent Element-Free Galerkin (CEFG) method [52,53], which extends the ideas of SCNI and QCI to the framework of linear, quadratic, and cubic approximations based on the Hu–Washizu three-field variational principle. In this formulation, three independent fields are considered: the displacement u, an interpolated strain ε~, and an assumed Cauchy stress σ^. The main purpose of the method is enforcing the orthogonality condition as a discrete level
∫Ωg(I)σ^T:(ε−ε~)dΩ=0,which ensures that the Hu-Washizu weak form reduces to the standard Galerkin formulation. The CEFG method constructs the interpolated strain ε~ using corrected nodal derivatives, which are computed such that the orthogonality condition holds for each integration subdomain. In practice, this involves solving local projection problems over the integration cells. This is similar to SCNI and QCI, but generalized to higher-order approximations. The CEFG scheme naturally reduces to SCNI and the QCI formulations when using linear and quadratic polynomial bases, respectively. The underlying rationale of the CEFG is that enforcing the Hu–Washizu orthogonality condition allows the strain interpolation to be variationally consistent, independent of the order of approximation.
The integration schemes discussed thus far have been primarily developed and validated for second-order elliptic partial differential equations, where the weak form involves first-order derivatives. However, the requirement for variational consistency extends to higher-order elliptic systems [206,207]. The most notable example is the fourth-order elliptic equations governing thin-plate bending, where the Sub-domain Stabilized Conforming Integration (SSCI) proposed by Wang and Chen [208] represented a significant evolution of the SCNI concept. While standard SCNI ensures consistency for first-order derivatives, SSCI partitions the smoothing cells into triangular sub-domains to satisfy the integration constraints required for bending exactness [208,209]. This allows the EFG framework to maintain spatial stability and pass the pure bending test, demonstrating that stabilized integration principles can be successfully tailored for complex mechanical systems involving second-order derivatives in their weak formulation.
It should be noted that the aforementioned integration schemes have been primarily developed and validated in the framework of elliptic problems, such as linear elasticity and Poisson-type equations. The variational principles and consistency conditions underpinning these methods rely on the symmetric and positive-definite nature of the associated differential operators, which facilitates both the satisfaction of patch tests and the convergence of the numerical solution.
For problems beyond this class, including non-symmetric, advection-dominated, or multiphysics systems, the direct application of SCNI, QCI, or CEFG is generally limited. To overcome these restrictions, alternative projection-based integration schemes have been proposed. Among these, the Reproducing Kernel-based Gradient Smoothing Integration (RKSGI) [54] and the Projection Integration (PI) [57] methods stand out as flexible approaches capable of handling a broader spectrum of governing equations. In the RKGSI framework, a smoothed gradient of a nodal shape function φ(I)(x→) is constructed over non-overlapping integration cells as Ω≊∑l=1ncΩl, with the nodes as cell vertices. This smoothed gradient is obtained through a reproducing kernel approximation defined as
∇~φ(I)(x→)=∫Ωl(y→)qT(y→)a~(x→)∇φ(I)(y→)dΩ=a~(x→)∫Ωl(y→)q(y→)∇φ(I)(y→)dΩ,which allows the derivative of the field to be consistently projected over the nodal integration domain, preserving the compatibility with the discrete divergence. The coefficients vector a~(x→) is obtained from the reproducing conditions as a~(x→)=qT(x→)G−1, where
G=∫Ωl(y→)qT(y→)q(y→)dΩ.
According to this, the smoothed gradient is given by
∇~φ(I)(x→)=qT(x→)G−1∫Ωl(y→)q(y→)∇φ(I)(y→)dΩ,which after integration by parts yields
∇~φ(I)(x→)=qT(x→)G−1[∫Γl(y→)q(y→)φ(I)(y→)n^dΓ−∫Ωl(y→)∇q(y→)φ(I)(y→)dΩ,].
It is worth noting that the smoothed gradient constructed in Eq. (68) is specifically designed to satisfy the compatibility condition of Eq. (63) over each integration cell Ωl, rather than over the smoothing domains associated with the nodes as in conventional nodal integration schemes. The integration by parts transformation eliminates the need for explicitly computing the standard MLS derivatives, since the smoothed gradient is directly obtained from boundary and volumetric integrals involving only the shape function itself. This procedure yields a compatible gradient field that preserves the discrete divergence constraint while enabling the use of low-order quadrature rules, in a manner analogous to FEM. From a computational standpoint, the construction of the smoothed gradient proceeds locally within each integration cell Ωl, and is performed in a cell-wise manner as follows:
Identify the vertices x→(I) associated with the integration cell Ωl.
Compute the moment matrix G over Ωl, and evaluate the volumetric and boundary integrals in Eq. (68) for each node contributing to the cell.
Assemble the local smoothed gradient ∇~φ(I)(x→) at position x→ within the cell for all nodes I associated with Ωl, using Eq. (68).
As a result, each node may possess multiple cell-wise smoothed gradients, one per adjacent integration cell. This systematic construction ensures variational consistency and high numerical stability, while avoiding the drawbacks associated with the direct differentiation of MLS shape functions. Although the RKGSI procedure involves the construction and inversion of the moment matrix G and the evaluation of additional integrals within each integration cell, it still represents a noteworthy reduction of computational cost when compared with standard EFG formulations using high-order Gaussian quadrature. This is mainly due to the drastic reduction in the number of integration points required, while maintaining numerical accuracy and stability.
To overcome the aforementioned integration difficulties, Wang and Ren [57] proposed the projection integration (PI) method. This approach introduces a linear projection operator 𝒫:𝒱h→𝒲h that systematically approximates the meshfree shape functions φ(I)(x→)—which define the EFG approximation space 𝒱h—with in a consistent polynomial subspace 𝒲h. Based on standard FEM shape functions that fulfil the Kronecker delta property, the projected shape functions are constructed as
φ~(I)(x→)=∑J=1nnN(J)(x→)φ(I)(x→J),where N(J)(x→) are linear interpolating shape functions defined over the integration cell Ωl, and nn is the number of its vertices. The resulting φ~(I)(x→) provides a piecewise polynomial approximation of the original EFG shape function φ(I)(x→), preserving the local consistency within Ωl. The projected shape functions are then used to solve the weak form problem within the projected approximation space 𝒱~h, i.e.,
a(δu~h,u~h)=ℓ(δu~h)∀δu~h∈𝒱~0h,where u~h=∑I=1nφ~(I)(x→)u(I) denotes the projected trial function.
Although the weak form is integrated using the projected basis φ~(I), the resulting nodal parameters u(I) correspond to the original EFG approximation space 𝒱h. Hence, the final solution is reconstructed with the non-projected shape functions as in Eq. (6), preserving the variational consistency of the method. The projection operator 𝒫 possesses the property of projection consistency, ensuring that the Galerkin weak form remains variationally consistent when expressed in terms of the projected basis. Owing to the piecewise polynomial nature of φ~I(x→), all domain and boundary integrals can be accurately evaluated using traditional low-order Gauss quadrature rules without additional enforcement constraints. As a result, the number of sampling points required per integration cell can be drastically reduced to save computing time compared to conventional EFG formulations based on high-order Gauss integration. Unlike the RKGSI approach, the PI method does not require the evaluation of additional domain and boundary integrals within each integration cell, nor the computation and inversion of the polynomial moment matrix G for the construction of smoothed gradients. Consequently, it achieves a comparable level of accuracy at a substantially reduced computational cost.
Both the RKGSI and PI methods provide a more general and robust integration framework. Their formulations are cell-based and problem-independent, enabling their straightforward extension to complex multiphysics and transport phenomena problems [53,55–57,144]. The combination of inherent variational consistency and reduced integration cost has significantly enhanced the computational efficiency and applicability of EFG methods beyond conventional elliptical problems. Nevertheless, it should be noted that the PI scheme relies on finite element shape functions for the projection process. Therefore, it may inherit some of the mesh distortion sensitivities associated with standard FEM. In scenarios involving extremely large deformations, the geometric degradation of the background integration cells could potentially limit the accuracy and robustness of the projection. This is a factor that should be carefully considered when choosing between projection-based and strictly meshless integration strategies.
A comparison between the main features of the improved integration schemes discussed in this section is provided in Table 1.
Comparative summary of the main EFG improved integration schemes.
Method
Integration Domain/Principle
Gradient Computation
Consistency/Stability
Computational Cost
Remarks and Applicability
DNI [199]
Voronoi cell around each node; single-point integration
Direct MLS derivatives
Low consistency; fails patch test; unstable without stabilisation
Very low
Simplifies integration; no background mesh; inaccurate stresses near boundaries.
RDNI, NDNI, MDNI [200,201]
Same as DNI with additional stabilisation terms
Direct MLS derivatives + residual, Taylor or least-squares correction
Improved stability; partial linear consistency
Low
Alleviates DNI instability; still fails linear patch test; costly higher-order derivatives.
SCNI [202]
Nodal subdomains via conforming Voronoi tessellation (smoothing domains Ωg(I))
Smoothed gradient ∇~φ(I) via integration constraint
Variationally consistent for linear fields; passes linear patch test
Low
Eliminates DNI instability; limited to linear consistency; accuracy comparable to linear FEM.
QCI [203–205]
Same domains as SCNI; satisfies DAC and DDC conditions
Generalisation of SCNI/QCI; allows independent displacement, strain, and stress fields.
RKGSI
Non-overlapping integration cells Ωl (cell-based)
Smoothed gradients via reproducing kernel projection and inversion of moment matrix G
Variationally consistent; stable; compatible with discrete divergence
Moderate
Cell-wise integration; fewer integration points than standard EFG; suitable for general PDEs.
PI [57]
Integration over cells using polynomial projection 𝒫:𝒱h→𝒲h
No explicit gradient smoothing; projected shape functions φ~(I)
Projection consistency; variationally consistent weak form
Low
No matrix inversion or extra integrals; accurate with low-order Gauss rules; general and efficient for multiphysics problems.
Volumetric Locking and Advection-Dominated Problems
Although EFG formulations are well-suited for addressing a wide range of problems in computational mechanics, such mesh-free techniques can still exhibit numerical difficulties similar to those encountered in conventional mesh-based methods. Two of the most critical challenges to be overcome for their successful application to transport and multiphysics problems are volumetric locking and advection-dominated behaviour.
Volumetric Locking
The first attempt to address volumetric locking in EFG formulations was made by Dolbow and Belytschko [34] in the context of nearly incompressible linear elasticity. Starting from a mixed displacement–pressure variational principle, the authors derived a selective reduced integration (SRI) scheme specifically tailored for meshfree methods. In their formulation, the deviatoric part of the weak form is integrated using standard background quadrature, whereas the dilatational (volumetric) contribution is evaluated through nodal integration based on Voronoi cells. This mixed integration strategy effectively alleviates volumetric locking across the full practical range of support sizes, without introducing spurious pressure modes. The approach thus preserves the meshfree character of EFG while ensuring accuracy and stability in the nearly incompressible regime. Later, Huerta et al. [184] proposed an alternative strategy to deal with incompressibility in mesh-free formulations. It consisted in developing a pseudo-divergence-free (PDF) EFG method by introducing an interpolation space that approximately satisfies the divergence-free constraint at a given discretisation. This procedure converges to a truly divergence-free space as the nodal resolution increases, since it relies on diffuse derivatives conceived to construct modified polynomial basis functions. These approximations exhibit asymptotic divergence-free behaviour, while maintaining the standard Galerkin structure without increasing the computational cost. Through inf–sup tests and benchmark Stokes flow problems, the PDF EFG formulation demonstrated improved stability compared to both standard EFG and mixed finite element schemes. In the framework of fluid dynamics, several authors have adopted the Characteristic-Based Split (CBS) formulation within the EFG framework. The CBS method can be regarded as a stabilised semi-implicit scheme that enhances the robustness of the numerical solution in convection-dominated and incompressible flow problems. Its key feature lies in the characteristic-based discretisation of the temporal derivative, which naturally stabilises the convective term while maintaining an optimal Galerkin spatial approximation. Moreover, the splitting of the governing equations introduces an inherent stabilisation of the pressure field. This allows the use of equal-order interpolation for velocity and pressure, circumventing the inf–sup condition. Accordingly, CBS-based EFG formulations effectively mitigate numerical instabilities and provide a reliable framework for simulating incompressible flows [102,103,115,210,211]. Although CBS-based formulations effectively enhance stability and circumvent the inf–sup condition, the achievement of a good performance is restricted to a narrow admissible range of the time-step size. To overcome this limitation, Wang and Ouyang [104] introduced the Element-Free Taylor–Galerkin method with Non-Splitting Decoupling (EFTG-NSD). In this approach, a Taylor–Galerkin discretisation imparts an inherent upwind effect that broadens the stable time-step range and improves robustness for steady flow problems.
An alternative approach was proposed by Álvarez-Hostos et al. [11,15,106,107,113], by developing EFG methods for the solution of steady incompressible Navier–Stokes equations through the implementation of standard penalty procedures originally formulated within FEM frameworks [5]. The consistent penalty method (CPM) and the reduced integration penalty method (RIPM) were adapted to EFG formulations, yielding stable and accurate velocity–pressure solutions without requiring additional stabilization schemes. Both formulations were shown to effectively alleviate volumetric locking and to produce smooth, oscillation-free pressure distributions across the domain. In the CPM-EFG formulation, incompressibility is enforced through a mixed two-field penalty approach analogous to that employed by Dolbow and Belytschko [34] in incompressible elasticity, whereas the RIPM-EFG formulation avoids the explicit construction of pressure shape functions by introducing a velocity-based penalty term to approximate the velocity divergence free constraint as ∇⋅v=−p/λ. This is conceived to achieve a weak form of the momentum equation which is entirely based on velocity [15,106,107]
∫Ωδvv˙dΩ+∫Ωδv⋅(v⋅∇v)dΩ+∫Ω∇δv:ν∇vdΩ+∫Ω∇⋅δvλ∇⋅vdΩ=∫Ωδv⋅bdΩ+∫Γδv⋅sdΓ,where the penalty term (fourth term on the left-hand side) is computed through one-point reduced integration and corresponds to the pressure work term −∫Ω∇⋅δvpdΩ in the standard Galerkin formulation. The divergence-free constraint is approximately imposed via an appropriate selection of the penalty coefficient λ. The magnitude of λ must be sufficiently high to ensure a virtually null divergence of the velocity field, while remaining bounded to prevent ill-conditioning of the global matrices. A robust penalty parameter is determined as λ=c12lavg−2(μ+ρUlavg+ρlavg2Δt), where lavg denotes the average nodal spacing and c1 is a positive constant, typically set to c1=100 [15,107]. This formulation is specifically designed to weight the viscous, advective, and transient scales within the fluid dynamics governing equations. For instance, in the steady-state analysis of the lid-driven square cavity at high Reynolds numbers—where pressure singularities arise at the upper corners—this adaptive scaling allows the divergence-free constraint to be strictly enforced. Under these conditions, the maximum local divergence is maintained at ‖∇⋅v‖≈10−6, whereas average values across the domain reach levels as low as 10−12, confirming the robustness of the constraint enforcement without compromising the system stability. This penalty-based approaches result in simpler yet robust formulation capable of delivering stable pressure fields even for small support sizes, where standard fully integrated EFG approaches exhibit severe locking. Moreover, Álvarez-Hostos et al. [106] demonstrated that the nodally integrated CPM-EFG and the RIPM-EFG schemes are theoretically equivalent.
These reduced-integration-based formulations proved capable of extending the range of stable support sizes and efficiently solving benchmark incompressible flow problems. Despite the aforementioned advantages, the stable computation of pressure in these reduced-integration formulations is achieved by evaluating it only at the reduced integration points. This strategy produces a discontinuous pressure field across the domain and limits the solution accuracy and continuity. To overcome this drawback, Álvarez-Hostos et al. [113] proposed an MLS-based reconstruction of the pressure field from the cell-wise discontinuous values. This reconstruction restores continuity and improves both accuracy and convergence. The continuous field exhibits a super-linear convergence rate close to 1.5, whereas the original discontinuous field converges linearly. Given its demonstrated robust performance and straightforward implementation, the EFG-RIPM formulation has been successfully applied to a variety of challenging problems, including the steady-state analysis of lid-driven square cavities at high Reynolds numbers [107], forced-convection heat transfer problems [113], transient fluid–structure interaction using immersed boundaries [15], capturing of bifurcation in benchmark problems such as the vortex shedding in flow past cylinders and Hopf bifurcation in lid-driven square cavities [15], and also mixed convection with solid–liquid phase change [11]. In contrast to CBS-based and EFTG-NSD formulations, the EFG-RIPM method does not introduce intrinsic restrictions on the time-step size. The only limitations can arise from the physical requirements of the problem, such as the need for capturing particular transient phenomena, such as the aforementioned Hopf bifurcations or Karman vortex shedding. As shown in Figs. 4 and 5, the EFG-RIPM formulation achieves accurate and stable velocity–pressure solutions even in demanding flow regimes. The method produces smooth, oscillation-free pressure distributions and continuous vorticity fields, making appropriate use of the inherent smoothness of EFG shape functions that enable the direct computation of continuous spatial derivatives.
Steady-state solution of a lid-driven square cavity flow for <inline-formula id="ieqn-320"><mml:math id="mml-ieqn-320"><mml:mtext>Re</mml:mtext><mml:mo>=</mml:mo><mml:mn>20000</mml:mn></mml:math></inline-formula>, obtained using the EFG-RIPM [<xref ref-type="bibr" rid="ref-107">107</xref>].
Vortex shedding in the transient numerical solution of a flow past a cylinder for <inline-formula id="ieqn-321"><mml:math id="mml-ieqn-321"><mml:mtext>Re</mml:mtext><mml:mo>=</mml:mo><mml:mn>200</mml:mn></mml:math></inline-formula>, obtained using the EFG-RIPM [<xref ref-type="bibr" rid="ref-15">15</xref>].
Zhang et al. [97] extended the variational multi-scale (VMS) concept of Hughes et al. [212] to the EFG framework by developing the Variational Multi-Scale Element-Free Galerkin (VMEFG) method, and its has been performed firstly in the context of Stokes problems. In this formulation, the velocity and weighting functions are decomposed into coarse and fine scales. This allows splitting the weak form into two subproblems, where the fine-scale problem is solved analytically by assuming bubble-type functions for the unresolved velocity field within each integration cell. In the VMEFG formulation, the velocity and weighting functions are decomposed into coarse- and fine-scale components, such that v=v¯+v′ and δv=δv¯+δv′. Substituting these decompositions into the standard weak form of the Stokes problem and exploiting linearity, two coupled sub-problems are obtained. From this point onwards, all variables without an overbar are assumed to refer to the coarse-scale components for notational simplicity. The resulting coarse-scale problem is
∫Ω∇δv:ν∇(v+v′)dΩ−∫Ω∇⋅δvpdΩ+∫Ωδp∇⋅(v+v′)dΩ=∫Ωδv⋅bdΩ+∫Γsδv⋅sdΓ,whereas the fine-scale problem restricted to each integration cell Ωcell reads
∫Ωcell∇δv′:ν∇v′dΩ=∫Ωcellδv′⋅rdΩ,where r=b−∇p+ν∇2v represents the residual of the coarse-scale momentum equation and ν denotes the kinematic viscosity. The quantities b, s and p in Eq. (72) are the field forces, surface load vector and pressure, respectively. Assuming that v′ and δv′ can be represented by bubble-type functions b1 and b2 within each cell, the fine-scale field can be expressed as
v′|Ωcell={b1[∫Ωcell(∇b1⋅∇b2I+∇b1⊗∇b2)dΩ]−1∫Ωcellb2dΩ}⏟τ⋅r=τ⋅r,where τ denotes the stabilization tensor. It is worth mentioning that the coarse problem residual has been considered constant within each integration cell Ωcell, whereby r is allowed to be outside the integral in Ωcell. The substitution of the fine-scale velocity (74) into the coarse-scale equations problem (72) yields the stabilised weak form
∫Ω∇δv:ν∇vdΩ−∫Ω∇⋅δvpdΩ+∫Ωδp∇⋅vdΩ−∫Ω(ν∇2δv+∇δp)⋅τ⋅rdΩ=∫Ωδv⋅bdΩ+∫Γsδv⋅sdΓin which the additional residual-based terms account for the unresolved sub-grid scales effects, in order to stabilise the pressure–velocity coupling and suppress spurious oscillations.
The bubble functions b1 and b2, used to approximate the fine-scale velocity field v′ and weighting function δv′ involved in the sub-problem (73), are defined in the reference domain of each background cell Ωcell. These functions are expressed in local coordinates ξ=(ξ,η,ζ) and constructed to vanish along the cell boundaries, i.e., bcell(ξ)=0 for ξ∈Γcell. This property confines the fine-scale contribution within each integration cell, preserving the inter-cell continuity of the coarse-scale field while enabling the analytical evaluation of Eq. (74). In practice, simple low-order polynomial bubbles are typically used to provide adequate subgrid resolution at minimal computational cost. The local coordinates are defined over the reference domains of each cell type, with ξ,η,ζ∈[−1,1] for quadrilateral and hexahedral cells, and ξ,η,ζ∈[0,1] satisfying ξ+η≤1 for triangular cells and ξ+η+ζ≤1 for tetrahedral cells. Common examples include:
2D rectangular cell: bcell(ξ,η)=(1−ξ2)(1−η2),
2D triangular cell: bcell(ξ,η)=27ξη(1−ξ−η),
3D hexahedral cell: bcell(ξ,η,ζ)=(1−ξ2)(1−η2)(1−ζ2),
3D tetrahedral cell: bcell(ξ,η,ζ)=256ξηζ(1−ξ−η−ζ).
Alternatively, Zhang et al. [98] modified the VMEFG formulation by introducing the so-called two-level EFG (TLEFG) method. In this approach, the fine-scale problem is solved numerically within each background integration cell using a standard EFG formulation. The locally computed fine-scale field is then substituted into the coarse-scale equations, effectively coupling both scales within a unified meshfree framework. This strategy does not rely on the use of bubble shape functions explicitly, and maintains consistency between the fine- and coarse-scale approximations. However, it also increases the computational cost and provides only a marginal improvement in overall accuracy compared to the original VMEFG formulation. The main concept of the TLEFG lies in its hierarchical structure, whereas its numerical performance does not show significant advantages over the use of well-designed bubble functions. In contrast, the variational multiscale EFG (VMEFG) method has found broader and more successful application across a wide range of flow problems characterised by the inf–sup stability constraint and volumetric locking. Its natural stabilising mechanism, derived from the explicit treatment of the unresolved scales, has proven effective in the numerical solution of incompressible Stokes flows coupled to advection–diffusion problems [152]. More recent studies have further demonstrated its theoretical soundness and versatility in the solution of Oseen [213] and Navier–Stokes [108,214] equations, which contributed to its wider adoption in practical simulations.
An alternative approach to address the incompressibility constraint in mesh-free formulations was proposed by Li and Dong [101], who developed a divergence-free element-free Galerkin (DFEFG) method for the Navier–Stokes equations. In this formulation, the velocity field is approximated using a divergence-free moving least-squares (DFMLS) scheme. This ensures that the incompressibility condition is satisfied at the basis level. Nevertheless, the global approximation is divergence-free only in an asymptotic sense due to the influence of the MLS weight functions and the local nature of the reconstruction.
For instance, the standard MLS approximation of a 2D velocity field v can be expressed as
v(x→)=(vx(x→)vy(x→))=(pT(x→)ax(x→)pT(x→)ay(x→)),where p(x→) is the polynomial basis vector and ax, ay are the local coefficient vectors. To enforce ∇⋅v=0 in the MLS framework, the coefficients must satisfy
∂pT∂xax+∂pT∂yay=0.
From this condition, a divergence-free basis qdiv(x→) can be constructed such that
v(x→)=qdiv(x→)a~,where q is a vector-valued basis whose components fully satisfy the divergence-free condition. For instance, in the case of a linear polynomial basis p(x→)=(1,x,y)T, the corresponding divergence-free basis reads
qdiv(x→)=(100xy01x−y0),a~=(ax1,ay1,ay2,ax2,ax3)T.
The substitution of (75) in the functional of Eq. (2) and its subsequent minimisation yields a divergence-free MLS approximation that guarantees ∇⋅v=0 at the basis level. However, as noted by Li and Dong [101], the global approximation is divergence-free only in an asymptotic sense. This is primarily because the derivation of the divergence-free basis qdiv assumes that the local coefficient vectors a~ are constant. In the MLS framework, these coefficients are actually functions of the spatial coordinates, and their gradients introduce a non-zero divergence residual that only vanishes as the nodal support decreases. Despite the use of this specialised basis, the formulation retains the pressure approximation as an explicit variable. This is a necessary choice to enforce the incompressibility constraint that is not fully captured by the asymptotic basis, and to ensure the fulfilment of the inf–sup (LBB) stability condition [215]. Numerical results presented in [101] confirm the stability and convergence of this approach, highlighting its efficiency in resolving the velocity-pressure coupling compared with standard pressure-stabilised EFG formulations. It is worth mentioning that the analysis of Li and Dong [101] was conducted using a Bubnov–Galerkin formulation—where the test functions for the momentum equation are drawn from the same constrained divergence-free space as the trial functions—and its application has been primarily focused on problems involving low to moderate Reynolds numbers.
Although so far limited to Stokes problems, it is worth mentioning other stabilisation strategies that have also been proposed within the EFG framework. For instance, several authors [66,100,216] added a stabilising weighted least-squares term to the Galerkin weak form of the momentum equations, in order to overcome the inf–sup condition instabilities. These additional terms play a role analogous to that of the subgrid-scale stabilization in the VMEFG, but is constructed empirically through a mesh-dependent parameter. This is typically proportional to hT2, where hT denotes the characteristic size of the local integration cell. Although such stabilization achieves satisfactory numerical performance and effectively suppresses pressure oscillations, the choice of the parameter remains heuristic and problem-dependent. Consequently, the VMEFG framework—where the stabilisation term arises naturally from the analytical solution of the fine-scale sub-problem—offers a more systematic and physically consistent formulation. As a feasible alternative, Zhang and Li [99] developed a generalised EFG (GEFG) method for the steady Stokes equations. This formulation exploits the MLS approximations’ partition unity property to extend the velocity field with additional nodal cover functions, expressed as linear combinations of the nodal influence polynomials. The resulting enriched approximation introduces auxiliary variables, leading to a system of higher dimensionality but enhanced stability and flexibility. In contrast to residual-based or subgrid-scale approaches, the GEFG method attains stability through kinematic enrichment of the trial and test spaces, offering an alternative route to circumvent the inf–sup condition without introducing explicit stabilisation parameters. Finally, the polynomial pressure projection (PPP) method has emerged as a particularly efficient spatial stabilization strategy. As demonstrated by Goh et al. [217], the PPP approach circumvents the inf-sup condition by projecting the pressure field onto a lower-order polynomial space within each integration cell. This introduces a stabilization term that suppresses pressure oscillations and eliminates volumetric locking without requiring high-order quadrature, bubble functions, or the kinematic enrichment of the approximation space. Its ability to pass the mixed patch test and ensure optimal convergence rates makes it a robust tool for modelling both incompressible Stokes flow and nearly incompressible solids.
Advection-Dominated Problems
The extension of stabilised formulations to advection-dominated problems was first addressed by Huerta and Fernández-Méndez [186], who proposed a time-accurate and consistently stabilised EFG framework. Such an approach highlighted a noteworthy advantage of the EFG method over standard finite element techniques: EFG shape functions naturally belong to a subspace of H2(Ω), allowing the computation of second-order spatial derivatives without any global reconstruction or additional degrees of freedom. This property enables the straightforward implementation of classical stabilisation schemes such as the SUPG, Galerkin/Least-Squares (GLS), and Subgrid-Scale (SGS) methods in a fully consistent manner. In contrast, conventional FEM often neglects second-derivative terms in the residual-based stabilisation. This omission reduces consistency and deteriorates convergence rates. In FEM, linear piecewise approximations produce discontinuous derivatives at element interfaces. Increasing the polynomial order does not solve this issue, since the resulting interpolation remains only C0-continuous. The finite element solution is therefore not in H2(Ω), and second derivatives such as the Laplacian cannot be accurately evaluated. As a result, residual-based stabilisation terms that depend on these derivatives lose their consistency. Constructing approximations with continuous derivatives requires either complex C1 elements or costly post-processing to recover smooth gradients. In contrast, EFG methods allow a complete representation of all residual terms within each integration cell. This yields consistently stabilised formulations without ad hoc corrections or problem-dependent parameters, while preserving both stability and the theoretical convergence order in strongly advective regimes.
Álvarez Hostos et al. [9] extended these concepts to nonlinear advection–diffusion problems, particularly to heat transfer with solid–liquid phase change. Their work showed that when small nodal support domains are used, the second-order derivative terms appearing in the full SUPG weak form can be omitted without compromising stability or accuracy. This is achieved by selecting support sizes of dm=dmaxΔl, where Δl is the nodal spacing and dmax=1.15 is a suitable support multiplier. Under these stabilising guidelines, the formulation produces smooth and stable results with negligible loss of consistency. The IMLS-based shape functions preserve the continuity of the primary variables and their gradients even for such small supports. A slight overlap between neighbouring support domains is sufficient to ensure gradients continuity in the moving local approximations, eliminating the need for any post-processing. Consequently, derived quantities such as heat flux in heat transfer problems [9,11,42], or vorticity and velocity gradients in fluid dynamics [11,15] can be directly computed in a consistent and continuous manner. These features also ensure that the SUPG formulation within the EFG framework preserves its optimal convergence behaviour even in advection-dominated regimes. This inherent smoothness and versatility of EFG methods open the door to more sophisticated couplings. For instance, a promising research direction would be the integration of the divergence-free bases proposed by Li and Dong [101] into the SUPG framework. Although the DFEFG implementation has been first implemented within a Bubnov–Galerkin scheme—suitable for moderate Reynolds numbers—its extension to the SUPG formulation would allow for the simultaneous treatment of mass conservation (via the DFMLS basis) and convective stability (via the modified test functions). This would be particularly beneficial for high-speed incompressible flows where both the incompressibility constraint and the non-linear advective terms pose significant numerical challenges.
In the context of coupled multiphysics, the SUPG stabilisation is consistently applied to both the momentum and energy equations. In general, the stabilised weak form is obtained by modifying the basis functions of the test space as φ~(I)=φ(I)+τ(v⋅∇φ(I)), whereas the trial space remains merely spanned by φ(I). This adaptation is not completely trivial in the EFG setting, as it requires a local meshless definition of the stabilisation parameter τ. For fluid dynamics, it is given by
τ=[(2‖v‖hloc)2+(2Δt)2+(12μρhloc2)2]−1/2,whereas for thermal problems it is computed as
τ=[(2‖v‖hloc)2+(2Δt)2+(12kρcphloc2)2]−1/2,where the local characteristic length hloc is derived from the IMLS gradients as hloc=2‖v‖(∑I=1n|v⋅∇φ(I)|)−1. A critical adaptation for phase-change problems involves the use of an effective specific heat, cpeff=cp−Hf(dgs/dT) in the computation of τ, to account for the latent heat release within the mushy region through the temperature-dependent solid fraction gs(T). Since the term dgs/dT can introduce numerical singularities, it is evaluated via the gradient relationship dgs/dT=(∇gs⋅∇T)/‖∇T‖2. This ensures that the stabilisation remains robust even when the advective transport of energy increases significantly during the solid-liquid transition.
Numerical results reported by Álvarez Hostos et al. [11] confirmed that the EFG-SUPG approach retains a second-order convergence rate in problems with large Péclet numbers. These results demonstrate that even with reduced support sizes, consistently stabilised EFG formulations maintain their theoretical convergence order while ensuring numerical stability in convection-dominated flows. The inherent smoothness of the IMLS approximations allows the accurate computation of advective gradients without artificial diffusion, providing a robust and efficient stabilisation strategy for nonlinear transport phenomena.
A clear example of the stabilising effect of the SUPG scheme for nonlinear advection–diffusion problems solved with the EFG formulation is depicted in Fig. 6. The plots compare the numerical solution of a one-dimensional solid–liquid phase-change problem obtained using both the SUPG and the standard Bubnov–Galerkin (BG) formulations. The results show that the SUPG approach effectively suppresses the non-physical oscillations that appear in the BG solution, which become increasingly pronounced as the Péclet number rises.
Solution of the advection–diffusion benchmark problem via the EFG method, using both the standard Bubnov-Galerkin and the stabilised SUPG formulations [<xref ref-type="bibr" rid="ref-11">11</xref>].
For more details concerning this nonlinear advection–diffusion problem and the corresponding analytical solution, readers are referred to a previous work of Álvarez Hostos et al. [42].
The variational multiscale EFG (VMEFG) formulation for advection–diffusion problems was first introduced by Xiang and Zhang [154]. In a similar fashion of the VMEFG developed by Zhang et al. [97] in the Stokes problem framework, the VMEFG for advection-diffusion problems involves a decomposition of both the field u and the corresponding weighting function δu into its coarse- and fine-scale components. Substituting these decompositions into the standard weak form of the advection–diffusion equation yields the coarse- and fine-scale problems
∫Ωδua⋅∇(u+u′)dΩ+∫Ω∇δu⋅k∇(u+u′)dΩ=∫δufdΩ+∫Γqδuq0dΓ,∫Ωcell∇δu′⋅k∇u′+δu′a⋅∇u′dΩ=∫Ωcellδu′rdΩ,where r=∇⋅(k∇u)−a⋅∇u+f. The quantities f and q0 in Eq. (82) are the source term and surface flux, respectively. The fine-scale problem (83) is solved locally within each integration cell using suitable bubble functions b1 and b2 associated with the trial and weighting functions, respectively. This leads to an explicit expression for the fine-scale variable:u′={b1[∫Ωcell(b2a⋅∇b1+∇b2⋅k∇b1)dΩ]−1∫Ωcellb2dΩ}⏟τr.
Substituting the above expression for u′ into the coarse-scale Eq. (82) yields the final VMEFG formulation:∫Ωδua⋅∇udΩ+∫Ω∇δu⋅k∇udΩ−∫Ω(∇⋅(k∇δu)+a⋅∇δu)τrdΩ=∫ΩδufdΩ+∫Γqδuq0dΓ.
The third term on the left-hand side of Eq. (85) introduces a residual-based multiscale stabilisation, which acts along the direction of advection while preserving consistency as the stabilisation effect vanishes for the exact solution. While this approach shares the fundamental residual-based philosophy of the classical SUPG method, the VMEFG provides a more rigorous variational foundation for the stabilisation magnitude. In SUPG, the stabilisation parameter τ is typically introduced through external heuristic correlations or parameter-dependent formulas. Conversely, the stabilising parameter τ in the VMEFG approach is directly derived from the variational residuals of the fine-scale problem through the integral expression in Eq. (84). This provides a parameter-free approach where the stabilisation is intrinsically determined by the local sub-grid scale behaviour, ensuring stability for high-Péclet-number flows without the need for problem-dependent adjustments. Nevertheless, the practical performance of VMEFG in suppressing oscillations is linked to the nodal resolution relative to the local Péclet number. In under-resolved scenarios, the multiscale operator ensures a monotonic solution, although at the expense of potential numerical diffusion to maintain stability in regions with steep gradients. Further developments of the VMEFG concept have been proposed to extend its applicability and efficiency. Zhang and Xiang [157] generalised the formulation to advection–diffusion–reaction problems, whereas Peddavarapu and Raghuraman [155] have used LME shape functions to allow the imposition of essential boundary conditions as prescribed nodal values. More recently, adaptive VMEFG schemes have been proposed based on polygonal influence domains constructed from background Delaunay triangulations [156]. These formulations enable a flexible adjustment of the nodal supports according to the local flow characteristics while maintaining numerical consistency. For detailed descriptions in this regard, authors are referred to [119,138]. In such approaches, the use of small overlapping supports allows the generation of shape functions that emulate the piecewise-continuous behaviour of FEM approximations yet preserve derivative continuity. This allows the omission of second-derivative terms in the multiscale stabilisation without compromising stability or accuracy—an effect analogous to that achieved by Álvarez Hostos et al. [11,15] when extending the SUPG technique to EFG formulations.
EFG Methods in Multi-Physics
The progressive enhancement of EFG methods has expanded their applicability beyond single-physics problems. The advances discussed so far in this work on numerical integration, imposition of essential boundary conditions, and stabilisation strategies have addressed most of the limitations that initially restricted EFG methods to linear or weakly coupled systems. As a result, EFG methods now provide a mature and versatile framework capable of describing highly coupled thermo-mechanical, thermo-fluid, and magneto-hydrodynamics processes with noteworthy accuracy and flexibility. These methodological developments have also solved some of the long-standing challenges that hindered the reliable coupling of distinct physical fields in the EFG framework. The accurate imposition of essential boundary conditions, the consistent treatment of material discontinuities, and the introduction of robust background integration schemes have greatly improved the stability and accuracy of EFG analyses. Combined with the advances achieved through variational multiscale [98,152,154] and residual-based formulations [66,100,216], these improvements have established a solid foundation for multi-physics simulations free from geometric constraints or interpolation inconsistencies. Moreover, the straightforward extension of classical stabilisation schemes such as SUPG to the EFG framework [9,11,15,42], together with the adoption of reduced integration strategies to alleviate volumetric locking and pressure instabilities [11,15,106,107,113], has enabled robust and accurate solutions of complex fluid dynamic problems and thermo-fluid coupling phenomena.
Over the last decades, EFG-based multi-physics formulations have been successfully applied to a broad range of coupled problems. Representative examples include thermo-mechanical coupling in linear [18] and non-linear solid mechanics [78,79], coupled heat transfer and fluid flow [110–112], also including phase change effects [11], magnetohydrodynamics [119,121,123,124], among other particular applications such as free surface flows [109], fluid-driven fracture mechanics [218], linear [116] and non-linear [118] porous media flow problems, efficient solutions for the Oseen problem [213] and phase-field based analysis in brittle fracture mechanics [125–128]. The flexibility of EFG methods facilitates the direct coupling of fields with strongly differing spatial or temporal scales, while their inherent smoothness ensures consistent computation of derivative-dependent quantities such as stresses, fluxes, or Lorentz forces across interfaces and evolving boundaries. The transfer of variables between domains with different discretisation or topology is straightforward in EFG methods owing to the support- and influence-domain mechanisms governing the approximation, which are not subjected to any prescribed partition or connectivity of the discretised domain. In contrast, such an operation in FEM frameworks requires geometric searches and element-wise projections, often relying on spatial search algorithms such as octree- [219] or k-d-tree-based [220] methods to locate the corresponding elements.
Some of the techniques previously discussed for solving the Stokes and Navier–Stokes equations by means of EFG formulations can already be regarded as multi-physics in nature, since these problems inherently involve the simultaneous approximation of two distinct but coupled fields. Both velocity and pressure are approximated, and stabilisation strategies to ensure compatibility and accuracy in incompressible flow regimes are required. The coupling between these variables via the incompressibility constraint constitutes a saddle-point problem, which exemplifies a fundamental challenge of multi-field interaction within the EFG framework. Such formulations, therefore, provide a natural starting point for extending the EFG method to more complex multi-physics scenarios. The EFG-based solutions depicted in Figs. 4 and 5 have already demonstrated the achievement of accurate and stable velocity-pressure coupling via the implementation of RIPM and SUPG techniques to deal with the divergence-free constraint and advection-dominated flow conditions, respectively. The VMEFG method has also been successfully tested in fluid dynamics problems under advection-dominated flow conditions, within benchmark cases such as the lid-driven square cavity flow and the backward-facing step flow [108]. Nevertheless, results for high Reynolds numbers (Re≥5000) in the lid-driven square cavity flow have exhibited an excess of viscous transport when compared with the well-known benchmark data of Ghia et al. [221]. The most refined discretisation used in the VMEFG-based solution of Hu et al. [108] consisted of a uniform Cartesian distribution of 257×257 nodes, which did not suffice to accurately capture the boundary layer structure. Álvarez-Hostos et al. [15] overcame these limitations by implementing a node clustering strategy originally developed for mesh-based formulations. This approach is based on the coordinate transformation proposed by Ding [222], which concentrates nodes in regions exhibiting steep velocity gradients—typically near solid walls. This approach demonstrates once again the straightforward extension of well-established procedures used in FEM to the EFG framework. The node clustering is given by the coordinate transformation
(x,y)=12+12tanh{[2(x′,y′)−1]ε}tanh(ε),where (x′,y′) denote the nodal positions in the reference uniform nodal distribution, and ε is the clustering control parameter governing the degree of node concentration near the boundaries. To ensure a consistent definition of the nodal influence domains under the transformed coordinates, Álvarez Hostos et al. [15] proposed the Jacobian-based scaling of the influence domain dimensions[lxly]=[εtanh(ε)cosh2[ε(2x′−1)]00εtanh(ε)cosh2[ε(2y′−1)]][lx′ly′],where lx′ and ly′ represent the uniform nodal spacing along the x′ and y′ axes in the reference homogeneous distribution. The combined use of this spatial transformation with the SUPG stabilisation scheme for advection-dominated transport and the RIPM to overcome the saddle-point problem arising from the velocity-pressure coupling allowed the achievement of stable and accurate solutions at high Reynolds numbers. It is worth emphasising that clustering nodes near regions with steep velocity gradients was essential to achieving such convergence behaviour, which would not be attainable under a uniform distribution of nodes. For instance, whereas a 257×257 uniform nodes distribution (with minimum spacing hmin=1/256=0.0039) did not suffice to resolve the boundary layer, the non-uniform distribution of only 61×61 nodes with ε=3 achieved comparable accuracy thanks to a local minimum spacing approximately seven times smaller (hmin≈0.00055). This result highlights the significance of both the nodes clustering near regions of high velocity gradients and the corresponding Jacobian-based scaling of the nodes’ influence domain, enabling high-resolution flow features to be captured with significantly fewer nodes than would be required in a uniform distribution. This clustering procedure has also been successfully implemented in other multi-physics problems, such as the implementation of immersed boundary techniques for the numerical solution of fluid-solid interaction problems [15] or the simultaneously developing non-isothermal flow in ducts [113]. Some of these clustered node distributions and the corresponding results are given in Figs. 7 and 8, depicting the accuracy achieved by comparison with benchmark solutions. Furthermore, the EFG method under the RIPM and SUPG formulation has also been successfully applied to multiphysics problems beyond simple geometries.
Clustered nodes distribution and computed velocity profiles corresponding to the numerical solution of the lid-driven square cavity flow at <inline-formula id="ieqn-391"><mml:math id="mml-ieqn-391"><mml:mtext>Re</mml:mtext><mml:mo>=</mml:mo><mml:mn>5000</mml:mn></mml:math></inline-formula>, including the comparison with benchmark results reported in the literature Wahba [<xref ref-type="bibr" rid="ref-223">223</xref>] (empty circles) and Hachem et al. [<xref ref-type="bibr" rid="ref-224">224</xref>] (empty squares). The benchmark results are also in full agreement with the solution of Ghia et al. [<xref ref-type="bibr" rid="ref-221">221</xref>].
Clustered nodes distribution and computed temperature profiles corresponding to the numerical solution of the simultaneous developing non-isothermal flow between parallel plates at <inline-formula id="ieqn-392"><mml:math id="mml-ieqn-392"><mml:mtext>Re</mml:mtext><mml:mo>=</mml:mo><mml:mn>800</mml:mn></mml:math></inline-formula> and different Prandlt numbers Pr, including the comparison with a benchmark analytical solution based on a Karman–Pohlhausen integral analysis [<xref ref-type="bibr" rid="ref-225">225</xref>]. The results correspond to the axial variation of the dimensionless centreline temperature.
Figs. 9 and 10 illustrate two representative examples: forced convection in a non-isothermal lid-driven semicircular cavity with a circular obstacle, and mixed convection with solid–liquid phase change in a direct chill casting (DCC) process.
Numerical solution of the non-isothermal lid-driven semicircular cavity with a circular obstacle for <inline-formula id="ieqn-393"><mml:math id="mml-ieqn-393"><mml:mtext>Re</mml:mtext><mml:mo>=</mml:mo><mml:mn>2000</mml:mn></mml:math></inline-formula> and <inline-formula id="ieqn-394"><mml:math id="mml-ieqn-394"><mml:mtext>Pe</mml:mtext><mml:mo>=</mml:mo><mml:mn>1000</mml:mn></mml:math></inline-formula> [<xref ref-type="bibr" rid="ref-15">15</xref>], obtained using the EFG-RIPM for the fluid dynamics analysis [<xref ref-type="bibr" rid="ref-113">113</xref>].
Numerical solution of the fully-coupled fluid flow-heat transfer problem with solid-liquid phase change in a realistic DCC geometry, using the EFG-RIPM for the fluid dynamics analysis and a deferred approach to solve the phase-change issues in the thermal problem [<xref ref-type="bibr" rid="ref-11">11</xref>].
In the particular case of coupled fluid flow–heat transfer problems with solid–liquid phase change, such as the DCC model of Fig. 10, EFG methods have proven remarkably effective when implemented through fixed-domain techniques. Unlike front-tracking or adaptive re-meshing—which can become computationally prohibitive in complex 3D geometries—fixed-domain approaches allow for a unified treatment of the entire domain [8,10,11]. To enforce the no-flow condition in solid regions, a Darcy penalisation term is incorporated into the momentum equation. The permeability κ is modelled via the Carman–Kozeny relation [11]:1κ=Cmushy(1−gl)2gl3+ε,where gl is a continuous, monotonically increasing function of temperature. It remains at zero (gl=0) for temperatures below solidus (T<Tsol), increases from gl=0 at T=Tsol to gl=1 at T=Tliq, and remains at unity (gl=1) for temperatures above the liquidus temperature (T>Tliq). This formulation allows for a gradual and stable transition of both velocity and temperature from the liquid to the solid phase through the mushy zone. Furthermore, the EFG framework enables a more robust treatment of latent heat phase change effects when they are incorporated as heat source terms, rather than relying on the effective specific heat method. The latter often suffers from poor convergence due to the marked discontinuous behaviour of dgl/dT in cpeff=cp+Hfdgl/dT [8,10,11]. Within this context, EFG methods are particularly well-suited for heat source formulations because they operate under a global weak form that lacks the element-wise geometric parametrisation inherent to mesh-based methods. While techniques like FEM can effectively handle the discontinuities of the effective specific heat capacity approach through phase-wise discontinuous integration schemes linked to element geometry, the globally smooth and continuously differentiable MLS approximations in EFG naturally facilitate the reconstruction of marked phase fraction gradients in the framework of heat source-based formulations. This allows the EFG method to capture the pronounced temperature and velocity gradients generated during the transition without requiring complex discontinuous integration or artificial regularisation. Consequently, the consistent transmission of momentum and thermal fluxes is maintained throughout the mushy zone, as demonstrated in the DCC analysis shown in Fig. 10.
All the aspects discussed so far clearly demonstrate how the straightforward extension of well-established numerical strategies—such as the Reduced Integration Penalty Method (RIPM) and the Streamline-Upwind Petrov–Galerkin (SUPG) scheme—to the context of EFG methods has enabled the accurate and stable solution of a wide spectrum of multiphysics problems. These include not only standard fluid-dynamics applications, but also fully coupled fluid flow and heat transfer analyses. This also encompasses highly nonlinear phenomena such as solid–liquid phase change and fluid–structure interaction. The demonstrated capability of the EFG framework to handle such challenging coupled processes confirms its robustness and versatility for the simulation of complex multiphysics systems, without the geometric or interpolation constraints inherent to mesh-based techniques.
The potential of VMEFG formulations to solve complex multiphysics problems has also been successfully demonstrated. It has proven effective in solving saddle-point issues arising from the velocity–pressure coupling in Stokes [97,108,152] and Darcy flows [116,118], while also providing stabilised solutions under advection-dominated transport conditions. Furthermore, the method has been extended to the coupling with other physical fields such as fluid flow and heat transfer [110–112]. The challenging case of magnetohydrodynamics has also been successfully addressed [119,121,123,124], where the interaction between the Navier–Stokes and Maxwell equations introduces Lorentz forces into the momentum balance. These developments highlight the robustness and generality of the VMEFG framework for accurately capturing complex, strongly coupled physical phenomena. Several remarkable and visually compelling results of complex multiphysics problems addressed through the VMEFG approach are now presented, demonstrating its ability to handle strongly coupled phenomena with high numerical stability. Figs. 11 and 12 depict the VMEFG-based solutions of natural convection problems obtained by Zhang et al. [110,111] in inclined square enclosures with an elliptic obstacle and triangular cavities with a zig-zag shaped bottom wall, respectively. These examples demonstrate the capability of the VMEFG formulation to accurately solve coupled multiphysics problems within irregular geometries.
Isotherms of the solution obtained via the VMEFG in an inclined square enclosure with an elliptic obstacle at Rayleigh number <inline-formula id="ieqn-407"><mml:math id="mml-ieqn-407"><mml:mtext>Ra</mml:mtext><mml:mo>=</mml:mo><mml:msup><mml:mn>10</mml:mn><mml:mn>6</mml:mn></mml:msup></mml:math></inline-formula>. This figure has been taken from the work of Zhang et al. [<xref ref-type="bibr" rid="ref-111">111</xref>], and the parameter <inline-formula id="ieqn-408"><mml:math id="mml-ieqn-408"><mml:mi>a</mml:mi></mml:math></inline-formula> represents the ratio between the areas of the elliptic obstacle and the square enclosure. The results are presented for different values of <inline-formula id="ieqn-409"><mml:math id="mml-ieqn-409"><mml:mi>a</mml:mi></mml:math></inline-formula>, also varying the inclination angle of the square cavity.
Isotherms and streamlines of the solution obtained via the VMEFG in a triangular cavity with a zig-zag shaped bottom wall, for Rayleigh numbers ranging <inline-formula id="ieqn-410"><mml:math id="mml-ieqn-410"><mml:mtext>Ra</mml:mtext><mml:mo>=</mml:mo><mml:msup><mml:mn>10</mml:mn><mml:mn>3</mml:mn></mml:msup></mml:math></inline-formula> to <inline-formula id="ieqn-411"><mml:math id="mml-ieqn-411"><mml:mtext>Ra</mml:mtext><mml:mo>=</mml:mo><mml:msup><mml:mn>10</mml:mn><mml:mn>6</mml:mn></mml:msup></mml:math></inline-formula>. This figure has been taken from the work of Zhang et al. [<xref ref-type="bibr" rid="ref-110">110</xref>].
The considered cases involve high Rayleigh numbers, defined as Ra=gβΔTL3να, which quantify the relative importance of buoyancy forces to viscous forces. Large Rayleigh numbers indicate a convection-dominated regime, where the transport of both momentum and thermal energy is primarily advective rather than diffusive. Fig. 13 depicts the VMEFG-based solution developed by Chen et al. [112] for the natural convection within a porous square enclosure with a wavy left wall, for Ra=104,106, and increasing numbers of wave oscillations on the heated boundary. These results correspond to a Darcy number of Da=κ/L2=10−2, where κ is the medium permeability and L the characteristic length.
Isotherms of the solution obtained via the VMEFG in a porous square enclosure with a wavy left wall, for Rayleigh numbers <inline-formula id="ieqn-416"><mml:math id="mml-ieqn-416"><mml:mtext>Ra</mml:mtext><mml:mo>=</mml:mo><mml:msup><mml:mn>10</mml:mn><mml:mn>4</mml:mn></mml:msup><mml:mo>,</mml:mo><mml:msup><mml:mn>10</mml:mn><mml:mn>6</mml:mn></mml:msup></mml:math></inline-formula> and <inline-formula id="ieqn-417"><mml:math id="mml-ieqn-417"><mml:mtext>Da</mml:mtext><mml:mo>=</mml:mo><mml:msup><mml:mn>10</mml:mn><mml:mrow><mml:mo>−</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:math></inline-formula>. This figure has been taken from the work of Chen et al. [<xref ref-type="bibr" rid="ref-112">112</xref>].
The results reported by Zhang et al. [110,111] and Chen et al. [112] highlight the remarkable stability of the VMEFG method, not only in the velocity–pressure coupling but also in stabilising the numerical solution of the highly advective transport processes characteristic of natural convection at elevated Rayleigh numbers as high as Ra=106 to provide oscillation-free solutions. At these scales, the transport of momentum and thermal energy is almost entirely advective, yet the VMEFG-based solutions maintain sharp and stable isotherms without the non-physical spurious oscillations typical of standard Galerkin approximations.
It is important to note that the stabilisation parameters derived from the solution of the fine-scale problems in VMEFG procedures ensure numerical stability regardless of the magnitude of advective transport in coupled fluid flow-heat transfer problems. However, the formulation may introduce significant numerical diffusion for node distributions that are not sufficiently refined near the walls to resolve the thin thermal and hydrodynamic boundary layers at elevated Ra. Although the solution remains stable and free of spurious oscillations, it may become over-diffusive and failing to accurately capture the steep gradients characteristic of highly advective regimes. Therefore, the effectiveness of the VMEFG in these challenging multiphysics problems relies on a balance between the intrinsic multiscale stabilisation and adequate nodal clustering near the walls. An additional step forward in multi-physics coupling has been achieved through the solution of the generalised Darcy–Forchheimer–Brinkman model within the EFG framework. This formulation extends the classical Navier–Stokes equations by incorporating the effects of viscous drag and inertial losses arising from the porous matrix resistance. The inclusion of the Darcy and Forchheimer terms introduces strong non-linearities and spatial parameter heterogeneity, as permeability, porosity, and inertial coefficients may vary across the domain. Moreover, the Brinkman diffusion term accounts for viscous shear within the porous structure, enabling a unified representation of free-flow and porous regions. These additional couplings pose significant numerical challenges associated with stability, interface continuity, and convergence, particularly in regions with steep permeability gradients.
A VMEFG-based solution developed by Zhang and Fan [124] for three-dimensional magnetohydrodynamics (MHD) flows is presented in Fig. 14, corresponding to a cubic domain subjected to an inclined magnetic field. The variational multiscale formulation effectively stabilises the coupling between the velocity and the induced magnetic field, ensuring robust convergence even at a Hartmann number of Ha=1000. At this regime, the Hartmann layers become extremely thin (δ∼Ha−1), posing a significant numerical challenge that typically requires a very high density of elements near the boundaries. Zhang and Fan [124] reported that the VMEFG framework can effectively capture the boundary layer physics without the intensive refinement at the walls, often required by mesh-based methods with C0 interpolations. According to the authors, the integration of the variational multiscale concept allows the fine-scale components to capture residual errors, naturally generating the stabilisation needed to suppress non-physical oscillations. While a proper nodal distribution remains essential, the smoothness of the MLS approximations, combined with this multiscale treatment, enables the method to maintain physical consistency and resolve the steep gradients of the Hartmann and Shercliff layers under strong MHD coupling [18,124].
VMEFG-based solution of a three-dimensional MHD flow in a cubic domain subjected to an inclined magnetic field (<inline-formula id="ieqn-423"><mml:math id="mml-ieqn-423"><mml:mtext>Ha</mml:mtext><mml:mo>=</mml:mo><mml:mn>1000</mml:mn></mml:math></inline-formula>). This figure has been taken from the work of Zhang et al. [<xref ref-type="bibr" rid="ref-124">124</xref>].
Beyond the field of transport phenomena, other multiphysics problems also merit attention. One illustrative example is the thermo–elastic analysis of orthotropic structures developed by Zhang et al. [18]. This problem involves the coupling between the elliptic subproblems of elasticity and heat conduction within orthotropic media. Fig. 15 shows the comparison of the von Mises stress distributions obtained in a connecting engine rod using both FEM and EFG formulations. The EFG-based thermo–mechanical coupling model exhibited excellent accuracy and robustness. Zhang et al. [18] also demonstrated the closer agreement with reference solutions obtained by EFG methods compared to FEM, while maintaining high precision under both regular and irregular nodal distributions. Although the solution of these problems does not require special stabilisation procedures, the underlying multiphysics is very relevant for structural applications. An appropriate example is the design of composite structures, where thermal deformation and stress can be effectively controlled by adjusting the fibre off-angle and orthotropic material parameters.
Comparison of Von Mises stress distributions in a connecting engine rod obtained with FEM and EFG formulations. This figure has been taken from the work of Zhang et al. [<xref ref-type="bibr" rid="ref-18">18</xref>].
The continuous evolution of EFG formulations enables their application to a wide variety of real multiphysics problems. The demonstrated capability of the method to handle the intricate coupling between distinct physical fields—ranging from thermo-fluid and magnetohydrodynamic interactions to thermo-elastic behaviour in anisotropic solids—confirms its robustness and generality. The inherent smoothness and mesh-independence of EFG approximations provide a unified computational framework capable of consistently solving derivative-dependent quantities. These attributes, together with the flexibility to incorporate stabilisation, node-adaptation, and coupling strategies, have transformed EFG methods into a powerful and versatile tool for the simulation of both transport and structural multiphysics phenomena.
Domain Splitting Techniques and Hybrid Strategies
Beyond the development of efficient numerical integration schemes, additional innovative strategies have been proposed to mitigate the high computational cost associated with EFG formulations. Among them, domain splitting techniques and hybrid approaches stand out as effective means to accelerate the search for neighbouring nodes, reduce the cost of shape function construction, and improve the overall efficiency of the numerical solution. These methods are conceived to retain the accuracy and smoothness inherent to EFG methods, while significantly enhancing the scalability and performance in large-scale analyses. This is ultimately intended to enable the application of EFG-based methods in more realistic and computationally demanding applied multiphysics problems, involving a large number of unknown field variables (degrees of freedom).
Domain Splitting Techniques
The domain splitting techniques are efficient computational strategies designed to alleviate the cost of meshfree approximations, particularly in problems with a significant number of unknown variables. These procedures significantly reduce the dimensionality of operations involved in the construction of shape functions and in the system of equations assembly.
The first splitting approach introduced in the EFG formulation is the DSM [62,63,67,226], which introduces a finite difference discretisation along the splitting direction in the strong form of the governing equations. This decomposes an n-dimensional problem into a set of (n−1)-dimensional coupled sub-problems, which are solved using standard EFG methods. This dimensional reduction leads to a remarkable improvement in computational efficiency, as the number of neighbour searches and matrix operations is drastically reduced without compromising accuracy or stability.
A further development in the framework of splitting techniques is the Dimension Splitting MLS (DS-MLS) method [65,66,227], which splits the EFG shape functions for an n-dimensional problem into a set of (n−1)-dimensional MLS approximations. In this approach, the physical domain Ω⊂Rn is decomposed into a sequence of one-dimensional slices according to
Ω=⋃K=1L−1Ω(K)×[x(K),x(K+1))∪Ω(L),where Ω(K)=Ω∩{x=x(K)} represents the (n−1)-dimensional subdivision surface along the splitting direction x. The position vector x→ is expressed as x→=(x,x→⊥), with x denoting the coordinate along the splitting direction and x→⊥ the set of transverse coordinates in the (n−1)-dimensional subspace. The field variable u(x→) with x→∈Rn is then approximated in two successive stages. (i) For a fixed point x→⊥ in the subdomain Ω(K), the field is interpolated along the splitting direction using one-dimensional interpolations I(K)(x)uh(x,x→⊥)=∑K=1nsI(K)(x)uh(x(K),x→⊥),where I(K)(x) can be constructed using any approximation fulfilling the Kronecker delta property. (ii) On each splitting surface Ω(K), the approximation uh(x(K),x→⊥) is given by (n−1)-dimensional MLS shape functions
uh(x(K),x→⊥)=∑I=1nkφ(I)(x→⊥)u(I)(x(K)),where φ(I)(x→⊥) denotes the MLS shape function defined over the (n−1)-dimensional subdivision surface Ω(K). By coupling both stages, the final approximation of the field is obtained as
uh(x→)=∑K=1ns∑I=1nkI(K)(x)φ(I)(x→⊥)u(I)(x(K)),where ns denotes the number of subdomains or splitting stages along the primary (split) direction, and nk is the number of neighbouring nodes considered within each (n−1)-dimensional MLS interpolation along the transverse domain Ω(K). This coupling effectively reduces the dimensionality of matrix operations during the construction of shape functions. The DS-MLS formulation thus preserves the full-dimensional representation of the physical problem while significantly enhancing numerical efficiency, enabling its application to large-scale and strongly coupled multiphysics problems.
While the DS-MLS method focuses on decomposing the shape function construction itself, the DSM introduces a dimensional reduction directly into the governing formulation. In this sense, the more classical DSM approach can be described considering a partial differential equation defined over an n-dimensional domain Ω⊂Rn(n=2,3)ℒ(u(x→))=f(x→),x→=(x,x→⊥)∈Ω,where ℒ(⋅) denotes a general linear differential operator, u(x→) is the unknown field function, and f(x→) is a prescribed source term. The boundary Γ=Γu∪Γq is partitioned into essential and natural boundaries, over which the following conditions hold:u(x→)=u¯(x→),x→∈Γu,q(x→)=q¯(x→),x→∈Γq,with Γu∩Γq=∅. A finite difference discretisation is introduced along the splitting direction x, yielding a sequence of (n−1)-dimensional problems at discrete planes x=x(K), K=1,2,…,L. By approximating derivatives along x with finite difference operators, the governing equation can be rewritten as
ℒ⊥(u(K)(x→⊥))=f(K)(x→⊥)+ℛx(u(K−1),u(K),u(K+1)),x→⊥∈Ω(K),where ℒ⊥ represents the restriction of ℒ to the (n−1)-dimensional transverse domain Ω(K)=Ω∩{x=x(K)}, ℛx denotes the residual terms involving finite differences along the x direction, and u(K)(x→⊥)=u(x(K),x→⊥). This procedure transforms the n-dimensional problem into a system of (n−1)-dimensional coupled equations. Each (n−1)-dimensional problem is then solved independently using EFG approximations over its corresponding Ω(K), whereas the coupling between neighbouring layers is introduced through the finite difference terms in ℛx. This dimensional decomposition significantly reduces the computational burden of full n-dimensional analyses, as matrix assembly and neighbour searching are confined to (n−1)-dimensional subdomains. The DSM therefore enables EFG formulations to tackle high-dimensional problems with enhanced efficiency, while preserving accuracy and stability across subdomain interfaces.
Although both DSM and DS-MLS have been mostly implemented in the solution of standard elliptic [65], parabolic [62], and hyperbolic [63,150] problems, their potential in multiphysics applications has begun to emerge through their successful implementation in the solution of Stokes [66,67] problems.
Hybrid EFG–FEM Approaches
The analogies between FEM and EFG methods have fostered the development of hybrid formulations that combine the strengths of both techniques. Such approaches consist in the application of EFG and in different regions of the problem domain, in the search for a balance between accuracy and computational efficiency. Typically, the computationally intensive EFG formulation is used only in those areas demanding higher numerical precision due to the expected emergence of steep gradients, singularities, or discontinuities, whereas less computational demanding FEM-based calculations are performed elsewhere in the domain [164,166,183]. Other works have exploited the hybrid EFG–FEM framework to facilitate the direct imposition of Dirichlet-type boundary conditions at nodal positions. This is usually achieved by introducing a strip of finite elements along the corresponding boundaries, which allows for a standard enforcement of essential boundary conditions maintaining the meshfree performance in the inner regions [162,228]. To ensure a smooth transition between the FEM and EFG discretisations, most hybrid EFG–FEM formulations make use of interface elements that incorporate ramp or blending functions [162,229]. These functions provide continuity across the coupling interface and guarantee consistency between the two approximation schemes. However, this technique does not succeed in providing a smooth coupling in the shape functions derivatives [19]. More recent contributions have succeeded in achieving a direct coupling between EFG and FEM regions, thereby eliminating the need for interface or coupling elements [165,166,183]. Ullah et al. [165,166,183] achieved this via an EFG formulation based on LME approximations, whose weak Kronecker delta property naturally enables direct coupling with FEM discretisations at the interface nodes. This approach has proven highly effective in large-deformation, nonlinear inelastic analyses, offering accurate results with reduced computational cost. A noteworthy aspect of the coupling procedure of Zahur et al. [165,166,183] is that the LME approximations provide a seamless EFG-FEM transition in both the shape functions and the corresponding derivatives, thus exhibiting a superior performance with respect to MLS approximations with interface elements and ramp functions. This aspect is depicted in Fig. 16, which presents a 1D comparison between MLS with interface elements and the plain LME coupling.
One-dimensional comparison of hybrid EFG–FEM coupling strategies. (<bold>a</bold>) MLS approximation with interface elements and ramp functions introduces discontinuities in the derivatives near the coupling region. (<bold>b</bold>) LME-based coupling achieves a smooth and consistent transition across the interface without coupling elements. These figures were taken from the Ph.D. Thesis of Ullah [<xref ref-type="bibr" rid="ref-183">183</xref>].
Zambrano-Carrillo et al. [19] also succeeded in achieving a direct coupling by using MK interpolations in the EFG regions, but this procedure also introduced oscillations in the fields obtained as gradients of the primary field variables. Zhang et al. [135] proposed an MLS-based coupling strategy in which the support size varies linearly and decreases toward the interface, allowing the nodal spacing to match that of the FEM mesh and recover the Kronecker delta property in the transition zone. This method was successfully applied to topology optimisation problems, further demonstrating the robustness and versatility of hybrid EFG–FEM formulations.
Hybrid EFG–FEM formulations offer several advantages, particularly when EFG approximations are restricted to regions requiring enhanced numerical accuracy. However, these classical hybrid strategies still rely on well-defined coupling interfaces where both discretisations share common nodes. This topological requirement limits flexibility and complicates the analysis of problems where the region demanding high resolution moves, changes shape, or cannot be predefined a priori. To overcome the need for such node-to-node matching, recent research has turned toward overset (chimera-type) techniques conceived to allow independently generated discretisations to overlap within the problem domain. Although overset strategies are extensively developed in mesh-based frameworks with successful applications ranging from simple potential problems [230,231] to more complex multiphysics scenarios [24,25,232,233], the implementation in mesh-free and mesh-less methods remains incipient. A first step in this direction was the overset improved element-free Galerkin (Ov-IEFG) proposed by Álvarez-Hostos et al. [41]. These authors used improved MLS approximations to construct the shape functions of the EFG formulation, so that the term Improved EFG (IEFG) has been used in this chimera-type approach. The inherent smoothness of IMLS shape functions and the corresponding derivatives enables a direct and robust transfer of field variables between the overlapping domains, which is performed iteratively through suitably defined immersed boundaries to yield a consistent coupling without the need for a prescribed topological relationship between the overlapping domains. Although Ov-IEFG exhibited naturally feasible coupling within EFG frameworks, the method still involves computationally demanding EFG computations in both domains. To address this limitation, Álvarez-Hostos et al. [11] subsequently proposed the overset EFG–FEM (Ov-IEFG-FEM) approach. This hybrid overset formulation combines a coarse finite element mesh used to discretise the full computational domain, with a fine cloud of EFG nodes placed only in the region requiring higher accuracy. The overlapping discretisations interact through a reciprocal exchange of nodal and derivative-based fields across immersed boundaries, maintaining consistency for any primary variable and its associated gradients. The resulting Ov-IEFG–FEM strategy retains the flexibility and smoothness of EFG approximations in critical areas, while leveraging the lower computational cost of FEM elsewhere.
The overlapping domains depicted in Fig. 17 can be used to provide a conceptual description of the Ov-IEFG-FEM, which consists in constructing a unified enriched solution from two overlapping discretisations: a coarse finite element mesh discretising the entire problem domain ΩFEM≊Ω, and a fine cloud of EFG nodes defining the patch domain ΩIEFG.
Representation of the problem domain for a numerical solution performed via the proposed Ov-IEFG-FEM [<xref ref-type="bibr" rid="ref-42">42</xref>].
The background FEM mesh provides a coarse global approximation of the field variable u, whereas the EFG patch enhances the solution in regions demanding increased numerical accuracy. The boundaries involved in the coupling are ΓIEFG (the external boundary of the patch) and the closed surface 𝒮IEFG→FEM, which encloses a set of background FEM nodes 𝒩enc. These two interfaces enable the reciprocal transfer of information between the overlapping domains. The FEM-based solution computed in ΩFEM is used to compute the field u¯FEM to be imposed on ΓIEFG, ensuring that the patch solution remains consistent with the background. The imposition of such a Dirichlet condition for ΩIEFG can be performed by methods such as penalty or Nitsche’s. Conversely, the refined solution computed in ΩIEFG is used to prescribe updated values of u at the nodes in 𝒩enc. This provides a local enrichment of the FEM solution within the region enclosed by 𝒮IEFG→FEM.
A crucial component of the coupling procedure is the high-order IMLS-based local reconstruction of the FEM solution used to evaluate u along ΓIEFG. This reconstruction is performed in a region Ωrec⊂ΩFEM, composed by the set of finite elements whose centres lie within the surface 𝒮rec. Using the nodes of Ωrec, the smooth and differentiable IMLS-based reconstruction of the FEM field is subsequently used for computing and imposing u¯FEM on ΓIEFG. The coupling between the solutions in ΩFEM and ΩIEFG is achieved through an iterative (recursive) imposition of the following constraints on the field variable u:u=u¯IEFGat x(h),h∈𝒩enc,u=u¯FEMon ΓIEFG,where u¯IEFG corresponds to the values of u computed in ΩIEFG via the IEFG technique and transferred to the enclosed background nodes, whereas u¯FEM denotes the reconstructed FEM-based values imposed on the patch boundary. This reciprocal exchange establishes a mutual dependence between the solutions obtained in the FEM background mesh and the IEFG patch. The patch solution inherits boundary information from the FEM discretisation, whereas the FEM solution is locally enriched by the higher-order IEFG approximation in the overlapping region. As a result, the Ov-IEFG-FEM method yields a unified solution u with enhanced accuracy only where required, without imposing any topological compatibility between the overlapping discretisations.
A noteworthy feature of the proposed Ov-IEFG-FEM method is that, similarly to the coupling strategy introduced by Ullah et al. [165,166,183], it provides a seamless transition in both the primitive field variable and its gradients across the overlapping domains. This method also dispenses with the restrictive requirement of a well-defined topological relationship between the overlapping discretisations, thereby offering substantially greater versatility and flexibility in the construction of hybrid numerical models. Illustrative examples of such positive features are given in Figs. 18 and 19 in the framework of transient heat conduction problems with moving heat sources and linear elasticity analysis, respectively.
Solution for the transient heat conduction problem of a concentrated moving heat source following a sinusoidal path, by using the Ov-IEFG-FEM [<xref ref-type="bibr" rid="ref-42">42</xref>].
Solution of the linear elasticity problem of an infinite plate with a centred hole, by using the Ov-IEFG-FEM [<xref ref-type="bibr" rid="ref-19">19</xref>].
The results of Fig. 18 exhibit a smooth and fully consistent transition of the temperature field across the overlapping domains, while simultaneously ensuring continuity in the associated heat flux related to temperature gradients through Fourier’s law. An analogous behaviour is observed in Fig. 19 that exhibits a seamless coupling not only in the displacement field, but also in the corresponding stress distribution linked to the displacement gradients through Hooke’s law. These results highlight the capability of the Ov-IEFG-FEM to preserve both primary variables and their derivative-based quantities across overset discretisations, ensuring high-quality solutions without the need for strict topological conformity between domains.
A fundamental feature of the Ov-IEFG–FEM is the robustness of its information transfer across the overlapping domains. To prevent the error accumulation often associated with standard interpolation, the method uses a constrained IMLS-based local reconstruction performed over a dedicated subdomain Ωrec⊂ΩFEM. Within this region, the background nodal values are strictly enforced in the IMLS functional via Lagrange multipliers [19,42]. This mathematical framework ensures a perfect nodal fit during the reconstruction, effectively reducing the error at the nodes to machine precision [42]. Furthermore, the formulation addresses the potential accuracy loss due to support truncation near the interfaces through two synergistic strategies. First, the reconstruction subdomain Ωrec is defined by a closed surface 𝒮rec that encloses the patch boundary ΓIEFG. This ensures that the coupling interface remains well-within the influence of the reconstruction nodes, providing sufficient nodal support to preserve the C1 continuity of the gradients. Second, a geometrical distance criterion is implemented [11,24,41,231], typically maintaining a separation of 1.5 to 3 times the average background element size between the internal ΓIEFG and external 𝒮IEFG→FEM embedded boundaries [19,42]. This spatial buffer prevents numerical instabilities and ensures that the IMLS shape functions possess complete nodal supports, preserving the smoothness of the solution. Finally, the consistency between the overlapping discretisations is ensured by a reciprocal iterative algorithm performed at each time step. This iterative process forces the patch to inherit consistent boundary information from the FEM background, while the latter is locally enriched by the high-order IEFG approximation.
This combination of perfect-fitting reconstruction and buffered embedded boundaries enables the seamless transition of both primary variables and derivative-based fields depicted in Figs. 18 and 19. The Ov-IEFG–FEM was initially developed using IMLS approximations to construct the EFG shape functions within the patch domain, leveraging their inherent filtering properties and high-order smoothness. This choice is particularly effective for overset coupling, as IMLS provides stable gradient transitions even when reconstructing fields from relatively coarse background FEM meshes. Nevertheless, other approximation schemes could also be considered within this hybrid framework. For instance, directly interpolating schemes such as MK or LME approximations—which possess a weak Kronecker delta property at the boundaries—could potentially simplify the imposition of coupling conditions. Although MK offers exact interpolation, its polynomial-based correlation functions may introduce numerical oscillations (Gibbs-like phenomena) across the domain when dealing with steep gradients or coarse background meshes. In contrast, LME approximations provide a more robust alternative since they are strictly non-negative. Additionally, LME satisfies the Kronecker delta property on the boundaries, ensuring stable interpolation at the interfaces without the oscillatory behaviour of MK. However, the computational cost of LME is significantly higher than that of IMLS, as it requires the solution of a non-linear optimisation problem via Newton-Raphson iterations for each evaluation point. Consequently, the selection of the approximation scheme remains a critical design choice. The trade-off between the exactness of the nodal fit, the overall smoothness, and the computational overhead of the recursive information transfer must be carefully balanced. In the particular case of a concentrated moving heat source followed by the moving patch ΩIEFG, the Ov-IEFG-FEM maintains numerical stability via an Arbitrary Lagrangian-Eulerian (ALE) description. Within this framework, an advective term is incorporated into the energy balance of the patch domain to account for the relative motion between the moving nodes and the fixed background mesh [41,42]. This term effectively handles the thermal flux entering and leaving the patch domain, ensuring that the high-gradient region is correctly transported without numerical lag or instabilities. The coupling stability is further reinforced by the high-order IMLS-based reconstruction with perfect nodal fit of the background solution via Lagrange multipliers, and the already explained geometrical distance criterion between the coupling interfaces (ΓIEFG and 𝒮IEFG→FEM). This spatial buffer prevents numerical oscillations when the high-gradient region enters or leaves specific portions of the global domain, preserving the C1 continuity of the thermal fluxes depicted in Fig. 18.
The motivation underlying both splitting techniques and hybrid EFG–FEM approaches lies on the search for computational efficiency without compromising accuracy. These strategies make use of the superior smoothness and high-order approximation capabilities of EFG methods only in regions—or along directions—where enhanced numerical precision is truly required, while relying on the lower-cost FEM elsewhere. These approaches provide an improved balance between accuracy and computational cost, making them particularly attractive for large-scale or multi-physics simulations. It is also worth noting that other hybrid approaches have been developed with promising results. For instance, the coupling of EFG with the Boundary Element Method (BEM) has been successfully implemented for elasticity problems [234], leveraging the boundary-only discretisation of BEM with the domain flexibility of EFG. The coupled EFG-finite strip method proposed by Mousavi et al. [235] for buclking analysis. Furthermore, a significant recent advancement is the consistent isogeometric–meshfree coupling. This hybrid framework integrates the geometry exact property of Isogeometric Analysis (IGA) with the discretisation flexibility of EFG methods. By establishing an equivalence between the IGA basis functions and the MLS approximations through reproducing conditions [236,237], this approach enables seamless adaptive refinement in regions of interest—such as areas with high temperature gradients or stress concentrations—while maintaining an exact geometric representation of the domain. This IGA-EFG coupling has proven particularly effective for heat-transfer simulations in anisotropic structures [237] and the limit analysis of cracked components [236].
Applications in Mechanics and Transport Phenomena
The ongoing improvement of EFG methods to allow their implementation in the solution of multiphysics problems has introduced a great versatility for these techniques, enabling a broad spectrum of applications in mechanics and transport phenomena. This has been particularly noteworthy in areas where evolving domains, marked nonlinearities and multiphysics coupling present substantial challenges for conventional mesh-based discretisations. A main area of application is found in manufacturing processes, which inherently involve complex thermal, mechanical and thermo-mechanical interactions. Such challenging applications include solid↔liquid phase change phenomena in the thermal [9,69,72,73] and thermo-mechanical [72,78] analysis of casting processes, numerical simulation of arc [41,238,239] and friction stir [240–242] welding, moving heat sources in anisotropic media [243] and additive manufacturing [42,244], and also thermo-mechanically driven forming operations [16,81]. The mesh-independence of higher-order approximations in EFG methods greatly facilitates the accurate resolution of steep gradients, moving interfaces and path-dependent material behaviour. EFG methods have also played an important role in advancing computational fracture mechanics, especially within the framework of phase-field formulations for brittle and quasi-brittle fracture [125–128]. The higher-order continuity afforded by MLS-based shape functions enhances the regularisation of the crack surface density field, enabling efficient and accurate coupling between displacement and phase-field variables. Moreover, the capability to introduce adaptive refinement without re-meshing provides a natural setting for resolving evolving cracks and complex fracture patterns. Beyond mechanical and manufacturing applications, the EFG method has been adopted for a wide range of problems governed by coupled transport and reaction phenomena. Examples include tumour-induced angiogenesis described by nonlinear reaction–diffusion to model the endothelial cells interaction, biochemical factors and extracellular matrix components [146]. EFG methods have also been successfully applied in groundwater hydrology to model coupled flow and multispecies reactive transport in unconfined aquifers, offering improved numerical stability and reduced sensitivity to dispersion errors relative to classical FDM and FEM schemes [245]. These applications highlight the potential of EFG methods to deal with heterogeneous media, sharp fronts, and spatially or temporally varying transport parameters. These applications illustrate the robustnes, flexibility, and accuracy of the EFG methods in the solution of complex applied problems in mechanics and transport phenomena.
After the seminary work of Álvarez-Hostos et al. [73] proposing the EFG-based solution of the heat transfer problem involved in a continuous casting process under developed flow conditions, this numerical technique has been widely implemented in metal manufacturing processes involving solid↔liquid phase change effects. That work included validation of the predicted solidified-shell growth through comparison with other numerical models and with experimental data reported in the literature, as well as a comprehensive parametric analysis examining the correspondence between casting parameters and process performance. Altogether, it placed the potential of EFG methods for applied thermal analysis in continuous casting into an appropriate perspective. The good agreement between the EFG predictions and both numerical and experimental results reported in the literature is illustrated in Fig. 20, demonstrating the reliability of this numerical technique for such applications.
Comparison of the inverse model and experimental values of shell growth reported by (<bold>a</bold>) Tang et al. [<xref ref-type="bibr" rid="ref-246">246</xref>] and (<bold>b</bold>) Zhang et al. [<xref ref-type="bibr" rid="ref-247">247</xref>] with the shell-growth predictions obtained from the EFG-based solution developed by Álvarez-Hostos et al. [<xref ref-type="bibr" rid="ref-73">73</xref>].
Given the reliability and feasibility demonstrated in that study regarding the application of EFG methods to the thermal analysis of continuous casting processes, subsequent developments and improvements were introduced to solve increasingly complex problems in this context. Some of the achieved results are depicted in Fig. 21, which include: (a) Comparison of the shell growth predicted via an EFG-based model for a continuous casting process under a pseudo-transient moving cross-section slice approach with the experimental measurements [69]; (b) air gap predictions from a EFG-based thermo-mechanical analysis of a continuous casting process, compared with the FEM-based solution [79]; (c) domain growth and temperature distribution computed in the 3-D moving boundary problem concerning the start-up stage of the direct chill casting of an aluminium alloy slab [10]; (d) analysis on the non-uniform shell growth during the continuous casting of a round billet, when considering the real heterogeneous surface cooling behaviour measured from on-line process data [70]. Air-gap evolution, such as that of Fig. 21b during continuous casting processes, introduces a variable thermal resistance at the metal–mould interface, which is also the main source of non-uniform heat flux in more realistic models such as that of Fig. 21d. In the EFG-based thermo-mechanical analysis [78,79], this is handled by a temperature- and gap-dependent heat transfer coefficient.
Some results obtained in the thermal and thermo-mechanical analysis of both continuous casting and direct chill casting process, using EFG methods. (<bold>a</bold>) Shell growth prediction via a pseudo-transient moving cross slice approach [<xref ref-type="bibr" rid="ref-69">69</xref>]. (<bold>b</bold>) Prediction of air-gap considering the path-dependent viscoplastic flow in the thermo-mechanical analysis [<xref ref-type="bibr" rid="ref-79">79</xref>]. (<bold>c</bold>) Solution of the non-linear thermal problem with
moving boundary during the start-up stage of a direct
chill casting process [<xref ref-type="bibr" rid="ref-10">10</xref>]. (<bold>d</bold>) Non uniform shell growth over a cross-section slice during the continuous
casting of a round billet, due to the heterogeneous surface cooling
[<xref ref-type="bibr" rid="ref-70">70</xref>]
It is important to note that the air-gap development is a continuous physical process (as shown in Fig. 20b), whereby the transition in thermal contact conditions does not involve mathematical discontinuities but rather steep gradients in the heat transfer response. The high-order smoothness of the MLS approximations is particularly advantageous in this context, as it enables a continuous representation of the evolving thermal resistance without the numerical noise or oscillations that can arise in C0 formulations when contact conditions change rapidly [113]. The coupling is enforced through a boundary heat flux term that adaptively updates based on the mechanical separation, ensuring a physically consistent transition across the interface.
It is worth mentioning that these applications have involved several improvements in the EFG implementation itself, allowing the achievement of reliable results in the solution of such challenging applied problems. Álvarez-Hostos et al. [78,79] adapted the return mapping algorithms in a small-strain kinematic framework to the EFG formulation, in order to properly predict the steel path-dependent viscoplastic flow during the continuous casting process. Álvarez-Hostos et al. [8] demonstrated that using the effective specific heat approach to treat solid↔liquid phase change phenomena within EFG formulations is generally unsuitable, as it commonly hinders convergence during the iterative solution of the non-linear system. To overcome these limitations, more robust schemes based on the source-term technique [8], mixed formulations [10], and deferred evaluation of the solid-fraction gradient [11] were proposed. These alternatives make more effective use of the continuity and smoothness intrinsic to EFG shape functions, thereby ensuring convergence to yield stable and accurate results in the vicinity of phase-change regions. Moreover, the latter scheme was successfully adapted to the non-linear multiphysics problem involving the coupling of fluid flow and heat transfer with solid–liquid phase change, and was applied specifically to a direct-chill casting process featuring a curved axisymmetric geometry (results shown in Fig. 10) [11].
The use of EFG methods proved particularly advantageous to solve the moving boundary problem concerning the start-up stage of direct-chill casting processes [10,72], as the inherent freedom to introduce nodes where needed allows a straightforward simulation of the evolving computational domain associated with the downward motion of the bottom block. As previously discussed in this manuscript, the support/influence domains mechanisms involved in the construction of shape functions for EFG methods also simplify the transfer of variables between domains with different discretisations. This has been particularly advantageous for the thermo-mechanical of continuous casting processes, since the transfer of information from the thermal analysis—defined over the entire problem domain—to the finer discretisation restricted to the solidified region to predict its viscoplastic flow is straightforward [79]. These advances in the thermal analysis of phase-change problems using EFG methods also enabled their application to other manufacturing processes, as illustrated by the following examples.
The thermal model presented in Fig. 22 for an arc-welding process of an S235JR structural steel was developed using the Ov-IEFG formulation, whereas the model shown in Fig. 23 for an additive-manufacturing process of AlSi10Mg was constructed using the hybrid Ov-IEFG-FEM approach. These examples constitute direct applications of the already discussed hybrid overset strategies, where a fine EFG patch is introduced and allowed to move together with the moving heat source. By concentrating the mesh-free approximation precisely in the region where steep thermal gradients, pronounced nonlinearities, and solid–liquid phase-change phenomena occur, the method efficiently resolves the localised physics without resorting to extreme mesh refinement or cumbersome adaptive procedures. Such adaptive strategies are computationally expensive in EFG methods, since they require repeated neighbour searches and the continuous reconstruction of approximation spaces as the heat source travels through the domain. In contrast, the overset EFG–FEM framework enables a smooth and robust relocation of the high-accuracy patch. This maintains the approximation quality while preserving computational efficiency, which makes these methods suitable for practical applications involving moving manufacturing heads or heat sources.
Thermal analysis of the arc-welding process of an S235JR structural steel via the Ov-IEFG technique [<xref ref-type="bibr" rid="ref-41">41</xref>].
Thermal analysis of the direct metal laser sintering process (metal additive manufacturing) of an AlSi10Mg allow via the Ov-IEFG-FEM [<xref ref-type="bibr" rid="ref-42">42</xref>].
Several studies have already reported experimental validations of numerical solutions based on EFG formulations, progressively consolidating EFG as a reliable computational tool for the multiphysics analysis of practical engineering problems. Three representative examples are included in Fig. 24 for illustration: (a) the prediction of metal folding in a hot axisymmetric forging process of a cylindrical specimen; (b) a side-by-side comparison between bead cross-sections and the simulated fusion zone in an arc-welding process; and (c) a comparison between the experimentally observed crack path and that predicted by a phase-field-based fracture mechanics model. The red circles in Fig. 24a indicate the lateral surface points of the cylindrical specimen that experience significant rotational motion due to pronounced frictional effects, ultimately leading to the formation of new contact between the specimen and the upper plate—the phenomenon known as metal folding. The EFG-based numerical solution predicted an increase of approximately 2 mm in the surface contact generated by metal folding, which showed excellent agreement with the experimental measurements reported by Puchi-Cabrera et al. [81]. The curved shape of the simulated fusion zone depicted in Fig. 24b exhibits an excellent agreement with the experimental observations, demonstrating the potential of EFG methods to accurately solve the phase change non-linearities during the arc welding process. The predicted crack growth obtained of the phase-field fracture model depicted in Fig. 24c also exhibits excellent agreement with experimental observations, providing a compelling example of a multiphysics application. In this framework, the mechanical response described by the elasticity equations is intrinsically coupled with the evolution equation governing the crack surface density function. Accordingly, the crack path, stiffness degradation, and energy dissipation emerge naturally from the coupled solution of both fields. It is worth noting that earlier applications of EFG in fracture mechanics were developed within the linear elastic fracture mechanics (LEFM) framework, where several strategies—such as the visibility criterion, diffraction methods, and transparent boundary approaches—were devised to capture the mechanical discontinuity introduced by the crack [94,248–250]. In contrast, the phase-field formulation combined with the smoothness of EFG approximations provides a more robust and unified treatment of crack initiation and propagation, enabling numerical predictions that closely reproduce the observed experimental results.
Experimental validations of numerical solutions based on EFG formulations in different manufacturing and structural applications. These figures have been taken from the works of Puchi-Cabrera et al. [<xref ref-type="bibr" rid="ref-81">81</xref>], Champagne and Pham [<xref ref-type="bibr" rid="ref-238">238</xref>], and Shao et al. [<xref ref-type="bibr" rid="ref-127">127</xref>]. (<bold>a</bold>) Prediction of metal folding in a hot axisymmetric forging process of a cylindrical specimen [<xref ref-type="bibr" rid="ref-81">81</xref>]. (<bold>b</bold>) Side-by-side comparison between bead cross-sections and the
simulated fusion zone in an arc-welding process [<xref ref-type="bibr" rid="ref-238">238</xref>]. (<bold>c</bold>) Comparison between the experimentally observed crack path and that
predicted by a phase-field-based fracture mechanics model [<xref ref-type="bibr" rid="ref-127">127</xref>].
The works of Shao et al. [125–128] on fracture mechanics analysis via phase-field models employ QCI schemes for the assembly of the stiffness matrices, offering a representative example of how enhanced integration techniques can improve the efficiency of multiphysics simulations within EFG frameworks.
The increasing number of studies demonstrating the close agreement between EFG-based numerical predictions and experimental observations has progressively positioned EFG as a reliable computational tool for the analysis of nonlinear and multiphysics applied problems in engineering. This growing evidence of accuracy and robustness has also encouraged its adoption in other advanced computational fields, most notably in topology optimisation. The smoothness of the EFG approximations and the flexibility of the discretisation offer clear advantages for handling evolving geometries, complex constraints, and strongly coupled physical phenomena. Early studies on topology optimisation using EFG methods as an analysis tool focused on defining the design variables (artificial densities) at each integration point, accompanied by extensive discussions regarding appropriate sensitivity filtering schemes to suppress chequerboard patterns analogous to those observed in FEM [134–137]. A major shift occurred with the work of Zhang et al. [130–132,138,251], who demonstrated that directly approximating the artificial density field via moving least-squares is sufficient to obtain optimal topologies essentially identical to their FEM counterparts, but with markedly sharper boundary profiles and without the need for any sensitivity filtering. This improvement stems from the ability of EFG to construct higher-order continuous shape functions, thereby producing a much smoother density field and eliminating intermediate-density artefacts. This finding placed EFG methods in a particularly favourable position within the topology-optimisation community and has since motivated a growing body of research employing EFG for increasingly sophisticated optimisation formulations—ranging from thermo-mechanical and transient heat-transfer problems to multi-material and anisotropic designs—demonstrating the strong potential of mesh-free approximations in complex design-driven, multiphysics settings. An illustrative example on this potential of EFG methods, demonstrated by Zhang et al. [130–132,138,251], is given in Fig. 25, showing the topology optimisation for minimisation of heat dissipation compliance in a CPU cooling fan made of isotropic material.
Topology optimisation for minimisation of heat dissipation compliance in a CPU cooling fan made of isotropic material. This figure has been taken from the work of Zhang et al. [<xref ref-type="bibr" rid="ref-130">130</xref>].
The EFG-based topology optimisation leads to periodic structures with markedly fewer intermediate-density regions and significantly smoother boundaries compared with the FEM counterparts, even in the absence of any filtering technique. The figure also includes a 3D-printed prototype manufactured according to the EFG-optimal design, together with the corresponding temperature distribution obtained from the numerical solution of the thermal problem. The results confirm that the optimised structure effectively mitigates thermal gradients across the component, preventing the formation of localised hot spots and demonstrating the practical relevance of the EFG-derived solution. The versatility of the approach proposed by Zhang et al. [130] has been found to be useful and very effective even in the design of metamaterials with extreme properties, such as structures with negative thermal expansion [252] or negative Poisson’s ratio [253].
Introducing FlexiGal
To support the growing interest in EFG and hybrid numerical strategies, an open-source Julia library FlexiGal.jl (Flexible-Galerkin) is currently being developed for the solution of partial differential equations using EFG, FEM, and mixed EFG–FEM formulations. Although it is not yet aspiring to be a full-scale simulation platform, FlexiGal is being conceived as a transparent, modular environment in which the computational foundations of EFG methods can be explored with minimal abstraction. Each component—background-mesh generation, influence-domain construction, numerical integration, trial-space definition, and operator assembly—is exposed to the user through a clear and concise syntax. Although FlexiGal is currently limited to scalar problems, support for vector fields is actively being incorporated. The current version already includes IMLS approximations, with additional approximation schemes planned for future releases. At this stage, the library is intended primarily as a gateway tool for researchers and students wishing to understand the algorithmic workflow of EFG methods. Its minimal but expressive design allows users to quickly prototype new ideas, test stabilisation procedures, or explore hybrid domain-coupling strategies.
The project is inspired by more mature frameworks such as Gridap.jl [254], which provides a highly sophisticated infrastructure for finite element discretisation. Gridap includes an extensive library of elements, advanced variational tools, and optional distributed-memory parallelism [255]. FlexiGal will not aim to replicate the wide variety of FEM tools available in Gridap, but it will instead include a small set of standard FEM spaces with the role of complementing and hybridising with EFG formulations within a unified framework.
To illustrate the typical workflow of FlexiGal, a minimal example is presented below. The script demonstrates the discretisation of a unit square Ω=[0,1]×[0,1], the construction of an EFG trial space, and its use in the solution of a steady advection-diffusion problem. The example highlights the clarity with which boundary conditions, integration rules, and variational operators can be expressed in the library. The model problem considered is the advection–diffusion equation inn Ω with velocity field v(x,y)=(−100(y−0.5),100(x−0.5)), and boundary conditions T|(0,y)=0,T|(1,y)=5, andT|(x,0)∪(x,1)=5x.
The results obtained for the scalar variable T and its gradient is depicted in Fig. 26. These results depict the smoothness achieved not only in the scalar variable T, but also in the corresponding gradients via the IMLS approximations used for the EFG-based solution.
Solution obtained for the advection-diffusion problem in a square cavity, using the EFG method implemented in FlexiGal.
Conclusions
Over the past three decades, Element-Free Galerkin (EFG) methods have progressed from an innovative alternative for linear solid mechanics into a versatile numerical technology capable of addressing increasingly complex engineering problems. Continuous improvements in approximation schemes, numerical integration, boundary treatment, and stabilisation techniques have constantly strengthened the method, overcoming many of the limitations that originally constrained EFG approaches. One of the most positive features of EFG formulations is the ability to produce smooth, high-quality fields without requiring post-processing. This feature has proven particularly valuable in applications where accurate gradients, fluxes, and interface behaviour are essential. As a result, EFG methods are able to achieve numerical predictions that exhibit excellent agreement with experimental observations, even in demanding non-linear and strongly coupled scenarios. Progress in efficient integration schemes has not only reduced the computational cost of EFG formulations but has also fostered the tackling of larger-scale problems with practical engineering relevance. Their ongoing incorporation into transport phenomena, fully coupled multiphysics systems, and problems beyond classical symmetric or elliptic settings represents a promising line of development that continues to expand the reach of EFG methods. Stabilisation approaches specifically designed for these numerical techniques have further enhanced reliability in fluid flow, transport phenomena, and convection-dominated regimes, consolidating EFG as a robust tool beyond classical structural mechanics.
These advances have naturally promoted the extension of EFG towards multiphysics applications. Currently, the method is used in thermo-mechanical problems, heat transfer with phase change, fluid–structure interactions, porous flow, magnetohydrodynamics, and other coupled processes where smooth field descriptions and accurate transfer of information between physical variables are essential. The flexibility of support-domain construction greatly facilitates data exchange between different regions of a domain, making EFG particularly suitable for systems involving multiple materials or evolving interfaces. Additionally, substantial progress has been made in reducing computational cost. Strategies based on dimension splitting techniques, hybrid EFG-FEM formulations, and, more recently, overset (chimera-type) couplings have demonstrated that EFG can be implemented in an efficient and fully consistent manner. The hybrid approaches allow EFG to be applied only where its advantages are most beneficial—such as near interfaces, singularities, or regions requiring high smoothness—while FEM is used elsewhere to reduce computational cost. The resulting frameworks offer remarkable flexibility for complex geometries and large-scale multi-resolution analyses.
The accumulated evidence from structural mechanics, heat transfer, fluid dynamics, and transport phenomena has positioned EFG as a reliable and predictive numerical method for non-linear and multiphysics engineering applications. Its capacity to deliver smooth, physically consistent fields has also opened new opportunities in topology optimisation. Recent results indicate that EFG-based optimal designs naturally present cleaner boundaries, fewer intermediate densities, and improved manufacturability, often without requiring sensitivity filters or auxiliary techniques. This places EFG formulations in a particularly promising position for the design of micro-architectured materials, thermomechanical devices, and components intended for additive manufacturing.
Future developments are still needed to extend the impact of EFG methods, where the design of efficient iterative solvers that can exploit the good conditioning of formulations without penalties or multipliers is still a major challenge. Progress in preconditioning tailored to EFG systems of equations will be essential. Another priority is the implementation of scalable parallel algorithms for distributed-memory architectures, where the non-local nature of EFG demands new strategies for neighbour searches and data exchange. Improvements in adaptive refinement and dynamic support management can further reduce computational cost. These developments will be crucial to applying EFG to large-scale and high-fidelity multiphysics simulations.
In summary, the continuous methodological refinements, the expansion into multiphysics environments, and the development of hybrid and overset couplings confirm the EFG method as a powerful, flexible, and evolving tool in computational mechanics. The advancements reviewed in this work demonstrate their growing potential for high-fidelity simulations, multi-physics, and optimisation-based design, indicating a promising future development and large-scale adoption.
The authors gratefully acknowledge the support from the National Scientific and Technical Research Council (CONICET) and the National Technological University (UTN) of Argentina. The authors would also like to thank ASACTEI for its contribution to this research.
Funding Statement
This work was supported by the Santa Fe Agency for Science, Technology and Innovation (ASACTEI), grant number IO-2025-428.
Author Contributions
Álvarez-Hostos Juan C.: writing—original draft, writing—review & editing, literature review, conceptualisation, formal analysis, visualization. Zambrano-Carrillo Javier A.: writing—review & editing, literature review, visualization, data curation. Sarache-Piña Alirio J.: writing—review & editing, literature review, data curation, visualization. All authors reviewed and approved the final version of the manuscript.
Availability of Data and Materials
Not applicable.
Ethics Approval
Not applicable.
Conflicts of Interest
The authors declare no conflicts of interest.
ReferencesZienkiewiczOC. Finite element method. 7th ed. Amsterdam, The Netherland: Elsevier Science and Technology; 2013.MoukalledF. The finite volume method in computational fluid dynamics. Berlin, Germany: Springer; 2016.ZienkiewiczOC, TaylorRL, FoxDD. The finite element method for solid and structural mechanics. 7th ed. Oxford, UK: Butterworth-Heinemann; 2014.ZienkiewiczOC, TaylorRL, NithiarasuP. The finite element method for fluid dynamics. 7th ed. Oxford, UK: Butterworth-Heinemann; 2014.ReddyJN, GartlingDK. Finite element method in heat transfer and fluid dynamics. Abingdon, UK: Taylor and Francis Group; 2010.NithiarasuP. Fundamentals of the finite element method for heat and mass transfer. 2nd ed. Hoboken, NJ, USA: Wiley; 2016.Bendsøe MP, Sigmund O.Topology optimization. 2nd ed. Berlin, Germany: Springer; 2004.Álvarez HostosJ, Gutierrez-ZambranoEA, Salazar-BoveJC, Puchi-CabreraES, BencomoAD. Solving heat conduction problems with phase-change under the heat source term approach and the element-free Galerkin formulation. Int Commun Heat Mass Transf. 2019;108(4):104321. doi:10.1016/j.icheatmasstransfer.2019.104321.Álvarez HostosJC, BencomoAD, Puchi-CabreraES, FachinottiVD, TournB, Salazar-BoveJC. Implementation of a standard stream-upwind stabilization scheme in the element-free Galerkin based solution of advection-dominated heat transfer problems during solidification in direct chill casting processes. Eng Anal Bound Elem. 2019;106(4):170–81. doi:10.1016/j.enganabound.2019.05.008.Álvarez HostosJC, TournBA, Zambrano-CarrilloJA, Sarache-PiñaAJ, FachinottiVD. Solving heat conduction problems in the start-up stage of direct chill casting processes via a temperature-enthalpy mixed formulation based on the improved element-free Galerkin method. Int J Heat Mass Transf. 2023;212(9–12):124231. doi:10.1016/j.ijheatmasstransfer.2023.124231.Álvarez HostosJC, TournBA, BencomoAD, MascottoM, Zambrano-CarrilloJA, Sarache-PiñaAJ. A deferred approach to include solid-liquid phase change effects in the solution of advection-conduction heat transfer problems via the improved element-free Galerkin method. Eng Anal Bound Elem. 2025;172(11):106110. doi:10.1016/j.enganabound.2024.106110.LiuG-R. Meshfree methods. 2nd ed. Boca Raton, FL, USA: CRC Press; 2010.LiX, DongH. The element-free Galerkin method for the nonlinear p-Laplacian equation. Comput Math Appl. 2018;75(7):2549–60. doi:10.1016/j.camwa.2017.12.019.LiX, LiS. Analyzing the nonlinear p-Laplacian problem with the improved element-free Galerkin method. Eng Anal Bound Elem. 2019;100:48–58. doi:10.1016/j.enganabound.2018.04.004.Álvarez HostosJC, CruchagaMA, FachinottiVD, Zambrano CarrilloJA, ZamoraE. A plausible extension of standard penalty, streamline upwind and immersed boundary techniques to the improved element-free Galerkin-based solution of incompressible Navier-Stokes equations. Comput Methods Appl Mech Eng. 2020;372(2):113380. doi:10.1016/j.cma.2020.113380.Álvarez HostosJC, BencomoAD, Puchi CabreraES, GuérinJ-D, DubarL. Modeling the viscoplastic flow behavior of a 20MnCr5 steel grade deformed under hot-working conditions, employing a meshless technique. Int J Plast. 2018;103(2):119–42. doi:10.1016/j.ijplas.2018.01.005.BourantasG, ZwickBF, JoldesGR, WittekA, MillerK. Simple and robust element-free Galerkin method with almost interpolating shape functions for finite deformation elasticity. Appl Math Model. 2021;96(3):284–303. doi:10.1016/j.apm.2021.03.007.ZhangJ-P, WangS-S, GongS-G, ZuoQ-S, HuH-Y. Thermo-mechanical coupling analysis of the orthotropic structures by using element-free Galerkin method. Eng Anal Bound Elem. 2019;101(2):198–213. doi:10.1016/j.enganabound.2019.01.011.Zambrano-CarrilloJA, Álvarez-HostosJC, SerebrinskyS, HuespeAE. Solving linear elasticity benchmark problems via the overset improved element-free Galerkin-finite element method. Finite Elem Anal Des. 2024;241:104247. doi:10.1016/j.finel.2024.104247.ZhangJ, ShenY, HuH, GongS, WuS, WangZ, et al.Transient heat transfer analysis of orthotropic materials considering phase change process based on element-free Galerkin method. Int Commun Heat Mass Transf. 2021;125(20):105295. doi:10.1016/j.icheatmasstransfer.2021.105295.ZhangJ-P, ZhouG-Q, GongS-G, WangS-S, HuS. Steady heat transfer analysis of orthotropic structure based on Element-Free Galerkin method. Int J Therm Sci. 2017;121(2–3):163–81. doi:10.1016/j.ijthermalsci.2017.06.024.SunZ, ZengZ, LiJ, ZhangX. An immersed multi-material arbitrary Lagrangian-Eulerian finite element method for fluid-structure-interaction problems. Comput Methods Appl Mech Eng. 2024;432(5):117398. doi:10.1016/j.cma.2024.117398.ChenF, ZhongW, WanD. Numerical investigation of the water entry of inclined cylinders using dynamic sliding mesh method. Ocean Eng. 2022;257(4):111525. doi:10.1016/j.oceaneng.2022.111525.StortiBA, AlbanesiAE, PeraltaI, StortiMA, FachinottiVD. On the performance of a Chimera-FEM implementation to treat moving heat sources and moving boundaries in time-dependent problems. Finite Elem Anal Des. 2022;208(3–4):103789. doi:10.1016/j.finel.2022.103789.LiM-J, ChenJ, LianY, XiongF, FangD. An efficient and high-fidelity local multi-mesh finite volume method for heat transfer and fluid flow problems in metal additive manufacturing. Comput Methods Appl Mech Eng. 2023;404:115828. doi:10.1016/j.cma.2022.115828.GonzálezFA, BustamanteJA, CruchagaMA, CelentanoDJ. Numerical study of flow past oscillatory square cylinders at low Reynolds number. Eur J Mech-B/Fluids. 2019;75(2):286–99. doi:10.1016/j.euromechflu.2018.10.017.ElgetiS, SauerlandH. Deforming fluid domains within the finite element method: five mesh-based tracking methods in comparison. Arch Comput Methods Eng. 2015;23(2):323–61. doi:10.1007/s11831-015-9143-2.ZhangS, GuillemotG, GandinC-A, BelletM. A partitioned two-step solution algorithm for concurrent fluid flow and stress-strain numerical simulation in solidification processes. Comput Methods Appl Mech Eng. 2019;356:294–324. doi:10.1016/j.cma.2019.07.006.TrungN, LiuGR. Smoothed finite element methods. Abingdon, UK: Taylor & Francis Group; 2016.BelytschkoT, LuYY, GuL. Element-free Galerkin methods. Int J Numer Methods Eng. 1994;37(2):229–56. doi:10.1002/nme.1620370205.LuYY, BelytschkoT, GuL. A new implementation of the element free Galerkin method. Comput Methods Appl Mech Eng. 1994;113(3–4):397–414. doi:10.1016/0045-7825(94)90056-6.KaljevicI, SaigalS. An improved element free Galerkin formulation. Int J Numer Methods Eng. 1997;40(16):2953–74. doi:10.1002/(SICI)1097-0207(19970830)40:16<2953::AID-NME201>3.0.CO;2-S.CordesLW, MoranB. Treatment of material discontinuity in the Element-Free Galerkin method. Comput Methods Appl Mech Eng. 1996;139(1–4):75–89. doi:10.1016/s0045-7825(96)01080-8.DolbowJ, BelytschkoT. Volumetric locking in the element free Galerkin method. Int J Numer Methods Eng. 1999;46(6):925–42. doi:10.1002/(sici)1097-0207(19991030)46:6<925::aid-nme729>3.0.co;2-y.BelytschkoT, KrongauzY, OrganD, FlemingM, KryslP. Meshless methods: an overview and recent developments. Comput Methods Appl Mech Eng. 1996;139(1–4):3–47. doi:10.1016/S0045-7825(96)01078-X.KryslP, BelytschkoT. ESFLIB: a library to compute the element free Galerkin shape functions. Comput Methods Appl Mech Eng. 2001;190(15–17):2181–205. doi:10.1016/S0045-7825(00)00229-2.DolbowJ, BelytschkoT. An introduction to programming the meshless Element Free Galerkin method. Arch Comput Methods Eng. 1998;5(3):207–41. doi:10.1007/bf02897874.BelytschkoT, KryslP, KrongauzY. A three-dimensional explicit element-free galerkin method. Int J Numer Methods Fluids. 1997;24(12):1253–70. doi:10.1002/(sici)1097-0363(199706)24:12<1253::aid-fld558>3.0.co;2-z.JuSH, HsuHH. Solving numerical difficulties for element-free Galerkin analyses. Comput Mech. 2013;53(2):273–81. doi:10.1007/s00466-013-0906-z.OlliffJ, AlfordB, SimkinsDC. Efficient searching in meshfree methods. Comput Mech. 2018;62(6):1461–83. doi:10.1007/s00466-018-1574-9.Álvarez HostosJC, StortiB, TournBA, FachinottiVD. Solving heat conduction problems with a moving heat source in arc welding processes via an overlapping nodes scheme based on the improved element-free Galerkin method. Int J Heat Mass Transf. 2022;192:122940. doi:10.1016/j.ijheatmasstransfer.2022.122940.Álvarez-HostosJC, UllahZ, StortiBA, TournBA, Zambrano-CarrilloJA. An overset improved element-free Galerkin-finite element method for the solution of transient heat conduction problems with concentrated moving heat sources. Comput Methods Appl Mech Eng. 2024;418(5–8):116574. doi:10.1016/j.cma.2023.116574.PecoC, MillánD, RosolenA, ArroyoM. Efficient implementation of Galerkin meshfree methods for large-scale problems with an emphasis on maximum entropy approximants. Comput Struct. 2015;150(1):52–62. doi:10.1016/j.compstruc.2014.12.005.BabuškaI, BanerjeeU, OsbornJE, LiQ. Quadrature for meshless methods. Int J Numer Methods Eng. 2008;76(9):1434–70. doi:10.1002/nme.2367.BabuškaI, BanerjeeU, OsbornJE, ZhangQ. Effect of numerical integration on meshless methods. Comput Methods Appl Mech Eng. 2009;198(37–40):2886–97. doi:10.1016/j.cma.2009.04.008.ZhouJX, WenJB, ZhangHY, ZhangL. A nodal integration and post-processing technique based on Voronoi diagram for Galerkin meshless methods. Comput Methods Appl Mech Eng. 2003;192(35–36):3831–43. doi:10.1016/s0045-7825(03)00376-1.LuoX-B, LiDM, LiuB. A comparison study of the efficiency and accuracy of IEFG in solving elasticity problems using different nodal integration schemes. Math Comput Simul. 2023;206(2):561–87. doi:10.1016/j.matcom.2022.12.002.Ortiz-BernardinA, RussoA, SukumarN. Consistent and stable meshfree Galerkin methods using the virtual element decomposition. Int J Numer Methods Eng. 2017;112(7):655–84. doi:10.1002/nme.5519.NagevadiyaB, VaghasiaB, RachchhN, BhoraniyaR. Galerkin meshfree methods: a review and mathematical implementation aspects. Int J Appl Comput Math. 2019;5(4):105. doi:10.1007/s40819-019-0665-4.DuanQ, BelytschkoT. Gradient and dilatational stabilizations for stress-point integration in the element-free Galerkin method. Int J Numer Methods Eng. 2008;77(6):776–98. doi:10.1002/nme.2432.ChenJ‐S, HillmanM, RüterM. An arbitrary order variationally consistent integration for Galerkin meshfree methods. Int J Numer Methods Eng. 2013;95(5):387–418. doi:10.1002/nme.4512.DuanQ, GaoX, WangB, LiX, ZhangH, BelytschkoT, et al. Consistent element-free Galerkin method. Int J Numer Methods Eng. 2014;99(2):79–101. doi:10.1002/nme.4661.WangB, DuanQ, ZhaoM. Improved integration scheme for the second-order consistent element-free Galerkin method. Eng Anal Bound Elem. 2019;98:157–71. doi:10.1016/j.enganabound.2018.10.020.WangD, WuJ. An inherently consistent reproducing kernel gradient smoothing framework toward efficient Galerkin meshfree formulation with explicit quadrature. Comput Methods Appl Mech Eng. 2019;349:628–72. doi:10.1016/j.cma.2019.02.029.LiX. Element-free galerkin analysis of stokes problems using the reproducing kernel gradient smoothing integration. J Sci Comput. 2023;96(2):43. doi:10.1007/s10915-023-02273-8.LiX, LiS. Effect of an efficient numerical integration technique on the element-free Galerkin method. Appl Numer Math. 2023;193:204–25. doi:10.1016/j.apnum.2023.07.026.WangJ, RenX. A consistent projection integration for Galerkin meshfree methods. Comput Methods Appl Mech Eng. 2023;414:116143. doi:10.1016/j.cma.2023.116143.ZhangP, ZhangX, XiangH, SongL. A fast and stabilized meshless method for the convection-dominated convection-diffusion problems. Numer Heat Trans Part A Appl. 2016;70(4):420–31. doi:10.1080/10407782.2016.1177327.DehghanM, AbbaszadehM. Proper orthogonal decomposition variational multiscale element free Galerkin (POD-VMEFG) meshless method for solving incompressible Navier-Stokes equation. Comput Methods Appl Mech Eng. 2016;311(10):856–88. doi:10.1016/j.cma.2016.09.008.AbbaszadehM, DehghanM, NavonIM. A proper orthogonal decomposition variational multiscale meshless interpolating element-free Galerkin method for incompressible magnetohydrodynamics flow. Int J Numer Methods Fluids. 2020;92(10):1415–36. doi:10.1002/fld.4834.YangY, ChengH. Topology optimization for elasticity problem of isotropic and orthotropic materials based on the complex variable element-free galerkin method. Int J Appl Mech. 2024;16(10):181. doi:10.1142/s1758825124501163.HeW, ChengH. The complex variable dimension coupling method for 3D inhomogeneous transient heat conduction problems. Int J Numer Methods Eng. 2025;126(19):181. doi:10.1002/nme.70149.MengZJ, ChengH, MaLD, ChengYM. The hybrid element-free Galerkin method for three-dimensional wave propagation problems. Int J Numer Methods Eng. 2018;117(1):15–37. doi:10.1002/nme.5944.ChengH, PengMJ, ChengYM. A hybrid improved complex variable element-free Galerkin method for three-dimensional advection-diffusion problems. Eng Anal Bound Elem. 2018;97(2):39–54. doi:10.1016/j.enganabound.2018.09.007.SunF, WangJ, WeiQ, WuY. An improved meshless method based on the dimension splitting moving least-squares method for elasticity problems. Eng Anal Bound Elem. 2023;150:374–84. doi:10.1016/j.enganabound.2023.02.025.SunF, WangJ, XuY. An improved stabilized element-free Galerkin method for solving steady Stokes flow problems. Appl Math Comput. 2024;463:128346. doi:10.1016/j.amc.2023.128346.WangJ, SunF. A hybrid generalized interpolated element-free Galerkin method for Stokes problems. Eng Anal Bound Elem. 2020;111(3):88–100. doi:10.1016/j.enganabound.2019.11.002.YangH, HeY. Solving heat transfer problems with phase change via smoothed effective heat capacity and element-free Galerkin methods. Int Commun Heat Mass Transf. 2010;37(4):385–92. doi:10.1016/j.icheatmasstransfer.2009.12.002.Álvarez HostosJC, BencomoAD, Puchi CabreraES, Figueroa PoleoIM. A pseudo-transient heat transfer simulation of a continuous casting process, employing the element-free Galerkin method. Int J Cast Met Res. 2017;31(1):47–55. doi:10.1080/13640461.2017.1366002.CaiL, WangX, WangN, YaoM. Meshless method for nonuniform heat-transfer/solidification behavior of continuous casting round billet. Metallurgical Mater Trans B. 2019;51(1):236–46. doi:10.1007/s11663-019-01718-6.CaiL, WangX, YaoM, LiuY. Element-free galerkin meshless method on solidification behavior inside continuous casting mold. Metallurgical Mater Trans B. 2020;51(3):1113–26. doi:10.1007/s11663-020-01820-0.Álvarez HostosJC, BencomoAD, Puchi CabreraES. Element-free Galerkin formulation for solving transient heat transfer problems of direct chill casting processes. Can Metall Q. 2017;56(2):156–67. doi:10.1080/00084433.2017.1288882.Álvarez HostosJC, Puchi CabreraES, BencomoAD. Element-free galerkin formulation by moving least squares for internal energy balance in a continuous casting process. Steel Res Int. 2015;86(11):1403–18. doi:10.1002/srin.201400352.Álvarez-HostosJC, MascottoMR, BencomoAD, Sarache-PiñaAJ, FachinottiVD. A fully analytical solution for 1-D advection-conduction heat transfer problems with non-isothermal solid-liquid phase change. Int Commun Heat Mass Transf. 2024;153(4):107327. doi:10.1016/j.icheatmasstransfer.2024.107327.ZhouL, YangH, MaL, ZhangS, LiX, RenS, et al.On the static analysis of inhomogeneous magneto-electro-elastic plates in thermal environment via element-free Galerkin method. Eng Anal Bound Elem. 2022;134(1):539–52. doi:10.1016/j.enganabound.2021.11.002.AwasthiA, PantM. Thermoelastic fracture analysis in orthotropic media using optimized element free Galerkin algorithm. Mech Adv Mater Struct. 2022;31(2):271–96. doi:10.1080/15376494.2022.2114039.AwasthiA, PantM. A novel approach for thermoelastic fracture simulation in bimaterials using element-free galerkin method with improved interface modeling. Int J Comput Methods. 2024;21(9):2450022. doi:10.1142/s0219876224500221.Álvarez HostosJC, Puchi CabreraES, BencomoAD. Stress analysis of a continuous casting process, on the basis of the element-free galerkin formulation. Steel Res Int. 2016;88(2):197–218. doi:10.1002/srin.201600019.Álvarez HostosJC, BencomoAD, CabreraESP. Simple iterative procedure for the thermal-mechanical analysis of continuous casting processes, using the element-free Galerkin method. J Therm Stresses. 2017;41(2):160–81. doi:10.1080/01495739.2017.1389325.CaiL, WangX, WeiJ, YaoM, LiuY. Element-free galerkin method modeling of thermo-elastic-plastic behavior for continuous casting round billet. Metallurgical Mater Trans B. 2021;52(2):804–14. doi:10.1007/s11663-020-02054-w.Puchi-CabreraES, GuérinJ-D, La Barbera-SosaJG, Álvarez-HostosJC, MoreauP, DubarM, et al.Friction correction of austenite flow stress curves determined under axisymmetric compression conditions. Exp Mech. 2019;60(4):445–58. doi:10.1007/s11340-019-00492-5.BadnavaH, NourbakhshSH, PezeshkiM. An element-free Galerkin approach for rate- and temperature-dependent behavior of inelastic solids. Comput Part Mech. 2025;12(6):4591–623. doi:10.1007/s40571-025-01060-6.WuQ, LiuFB, ChengYM. The interpolating element-free Galerkin method for three-dimensional elastoplasticity problems. Eng Anal Bound Elem. 2020;115(1):156–67. doi:10.1016/j.enganabound.2020.03.009.YangY, ChengY, PengM, ChengH. The complex variable element-free Galerkin method based on the conjugate basis function for 3D elastoplastic problems. Eng Anal Bound Elem. 2025;179(4):106425. doi:10.1016/j.enganabound.2025.106425.WangYF, LuY, ChenL, PengMJ, ChengYM. The improved interpolating element-free Galerkin method based on nonsingular weight functions for three-dimensional elastoplastic problems. Eng Anal Bound Elem. 2025;172:106136. doi:10.1016/j.enganabound.2025.106136.WuQ, PengPP, ChengYM. The interpolating element-free Galerkin method for elastic large deformation problems. Sci China Technol Sci. 2020;64(2):364–74. doi:10.1007/s11431-019-1583-y.BadnavaH, LeeCH, NourbakhshSH, Refachinho de CamposPR. A stabilised total Lagrangian Element-Free Galerkin method for transient nonlinear solid dynamics. Comput Mech. 2024;75(1):327–55. doi:10.1007/s00466-024-02507-y.KumarSS, ShajiA, MuthuN. A two-field mixed formulation with scattered pressure node distribution in element-free Galerkin method for alleviating volumetric locking in hyperelastic materials. Acta Mech Sin. 2024;41(10):424446. doi:10.1007/s10409-024-24446-x.ChengH, HeW, ChengY. The improved complex variable element-free Galerkin method for 3D elastic large deformation problems. Eng Struct. 2026;354:122368. doi:10.1016/j.engstruct.2026.122368.LiuF, ChengY. The improved element-free Galerkin method based on the nonsingular weight functions for elastic large deformation problems. Int J Comput Mater Sci Eng. 2018;7(4):1850023. doi:10.1142/s2047684118500239.LiDM, TianL-R. Large deformation analysis of a gel using the complex variable element-free Galerkin method. Appl Math Model. 2018;61(8):484–97. doi:10.1016/j.apm.2018.04.004.FanW-L, GaoX-W, PengF, XuB-B. A total Lagrangian Galerkin free element method for finite deformation in hyperelastic materials. Appl Math Model. 2025;137:115740. doi:10.1016/j.apm.2024.115740.LuY, PengM, ChengY. The improved complex variable element-free Galerkin method for inhomogeneous large deformation of thermo-chemo-mechanical responsive hydrogels. Appl Math Model. 2025;140:115886. doi:10.1016/j.apm.2024.115886.PanJ-H, LiDM, LuoX-B, ZhuW. An enriched improved complex variable element-free Galerkin method for efficient fracture analysis of orthotropic materials. Theor Appl Fract Mech. 2022;121(3):103488. doi:10.1016/j.tafmec.2022.103488.ZhangJ, WuS, ZhangH, ZhaoL, ZuoZ, WuS. Topology optimization of orthotropic multi-material structures with length-scale control based on element-free Galerkin method. Eng Anal Bound Elem. 2024;163(10):578–92. doi:10.1016/j.enganabound.2024.03.031.PengL, ZhangZ, WeiD, TangP, MoG. Nonlinear analysis of corrugated core sandwich plates using the element-free galerkin method. Buildings. 2025;15(8):1235. doi:10.3390/buildings15081235.ZhangL, OuyangJ, ZhangX, ZhangW. On a multi-scale element-free Galerkin method for the Stokes problem. Appl Math Comput. 2008;203(2):745–53. doi:10.1016/j.amc.2008.05.081.ZhangL, OuyangJ, ZhangX-H. On a two-level element-free Galerkin method for incompressible fluid flow. Appl Numer Math. 2009;59(8):1894–904. doi:10.1016/j.apnum.2009.02.003.ZhangT, LiX. A generalized element-free Galerkin method for Stokes problem. Comput Mathem Appl. 2018;75(9):3127–38. doi:10.1016/j.camwa.2018.01.035.LiX, LiS. Meshless Galerkin analysis of the generalized Stokes problem. Comput Mathem Appl. 2023;144:164–81. doi:10.1016/j.camwa.2023.05.027.LiX, DongH. Analysis of a divergence-free element-free Galerkin method for the Navier-Stokes equations. Appl Numer Math. 2025;217(2):73–95. doi:10.1016/j.apnum.2025.06.002.ShamekhiA, SadeghyK. On the use of characteristic-based split meshfree method for solving flow problems. Int J Numer Methods Fluids. 2007;56(10):1885–907. doi:10.1002/fld.1529.YouC, QiuY, XuX, XuD. Numerical simulations of viscous flows using a meshless method. Int J Numer Methods Fluids. 2008;58(7):727–41. doi:10.1002/fld.1760.WangX, OuyangJ. Time-related element-free Taylor-Galerkin method with non-splitting decoupling process for incompressible steady flow. Int J Numer Methods Fluids. 2011;68(7):839–55. doi:10.1002/fld.2582.WangX, OuyangJ, SuJ, YangB. On the superiority of the mixed element free Galerkin method for solving the steady incompressible flow problems. Eng Anal Bound Elem. 2012;36(11):1618–30. doi:10.1016/j.enganabound.2012.05.006.Álvarez HostosJC, FachinottiVD, Sarache PiñaAJ, BencomoAD, Puchi CabreraES. Implementation of standard penalty procedures for the solution of incompressible Navier-Stokes equations, employing the element-free Galerkin method. Eng Anal Bound Elem. 2018;96:36–54. doi:10.1016/j.enganabound.2018.08.008.Álvarez HostosJC, Salazar BoveJC, CruchagaMA, FachinottiVD, Mujica AgelvisRA. Solving steady-state lid-driven square cavity flows at high Reynolds numbers via a coupled improved element-free Galerkin-reduced integration penalty method. Comput Mathem Appl. 2021;99(3):211–28. doi:10.1016/j.camwa.2021.08.013.HuG, LiR, ZhangX. A novel stabilized Galerkin meshless method for steady incompressible Navier-Stokes equations. Eng Anal Bound Elem. 2021;133:95–106. doi:10.1016/j.enganabound.2021.08.017.ZhangL, OuyangJ, JiangT, RuanC. Variational multiscale element free Galerkin method for the water wave problems. J Comput Phys. 2011;230(12):5045–60. doi:10.1016/j.jcp.2011.03.026.ZhangX, ZhangP. Meshless modeling of natural convection problems in non-rectangular cavity using the variational multiscale element free Galerkin method. Eng Anal Bound Elem. 2015;61:287–300. doi:10.1016/j.enganabound.2015.08.005.ZhangP, ZhangX, DengJ, SongL. A numerical study of natural convection in an inclined square enclosure with an elliptic cylinder using variational multiscale element free Galerkin method. Int J Heat Mass Transf. 2016;99:721–37. doi:10.1016/j.ijheatmasstransfer.2016.04.011.ChenJ, ZhangX, ZhangP, DengJ. Variational multiscale element free Galerkin method for natural convection with porous medium flow problems. Int J Heat Mass Transf. 2017;107:1014–27. doi:10.1016/j.ijheatmasstransfer.2016.11.008.Álvarez-HostosJC, TournB, Zambrano-CarrilloJA, Sarache-PiñaAJ, Rondón-SilvaLA, BencomoAD, et al.A simple staggered approach for comprehensive analysis of forced convection heat transfer using the improved element-free Galerkin-reduced integration penalty method to solve the fluid dynamics problem. Eng Anal Bound Elem. 2023;150(2):672–96. doi:10.1016/j.enganabound.2023.02.047.PengMJ, LiRX, ChengYM. Analyzing three-dimensional viscoelasticity problems via the improved element-free Galerkin (IEFG) method. Eng Anal Bound Elem. 2014;40(3):104–13. doi:10.1016/j.enganabound.2013.11.018.ZhangXH, OuyangJ, ZhangL. Characteristic based split (CBS) meshfree method modeling for viscoelastic flow. Eng Anal Bound Elem. 2010;34(2):163–72. doi:10.1016/j.enganabound.2009.08.001.ZhangT, LiX. Meshless analysis of Darcy flow with a variational multiscale interpolating element-free Galerkin method. Eng Anal Bound Elem. 2019;100:237–45. doi:10.1016/j.enganabound.2017.10.017.SamimiS, PakA. A novel three-dimensional element free Galerkin (EFG) code for simulating two-phase fluid flow in porous materials. Eng Anal Bound Elem. 2014;39(6):53–63. doi:10.1016/j.enganabound.2013.10.011.ZhangT, LiX. Variational multiscale interpolating element-free Galerkin method for the nonlinear Darcy-Forchheimer model. Comput Mathem Appl. 2020;79(2):363–77. doi:10.1016/j.camwa.2019.07.003.ZhangL, OuyangJ, ZhangX. The variational multiscale element free Galerkin method for MHD flows at high Hartmann numbers. Comput Phys Commun. 2013;184(4):1106–18. doi:10.1016/j.cpc.2012.12.002.JainS, BhargavaR. Numerical simulation of free convection of MHD non-Newtonian nanofluid within a square wavy enclosure using Meshfree method. Int J Comput Methods Eng Sci Mech. 2020;22(1):32–44. doi:10.1080/15502287.2020.1846096.JannesariZ, TatariM. Magnetohydrodynamics (MHD) simulation via an adaptive element free Galerkin method. Eng Comput. 2020;38(1):679–93. doi:10.1007/s00366-020-01079-8.LiX, LiS. Element-free Galerkin analysis of MHD duct flow problems at arbitrary and high Hartmann numbers. Eng Comput. 2024;40(5):3233–51. doi:10.1007/s00366-024-01969-1.LiX, DongH. Analysis of a stabilized element-free Galerkin method for magnetohydrodynamic flow at very large Hartmann numbers. Appl Math Comput. 2025;495(2):129334. doi:10.1016/j.amc.2025.129334.ZhangX, FanY. The variational multiscale element free Galerkin method for three-dimensional steady magnetohydrodynamics duct flows. J Comput Sci. 2025;91:102683. doi:10.1016/j.jocs.2025.102683.ShaoY, DuanQ, QiuS. Adaptive consistent element-free Galerkin method for phase-field model of brittle fracture. Comput Mech. 2019;64(3):741–67. doi:10.1007/s00466-019-01679-2.ShaoY, DuanQ, QiuS. Consistent element-free Galerkin method for three-dimensional crack propagation based on a phase-field model. Comput Mater Sci. 2020;179(2):109694. doi:10.1016/j.commatsci.2020.109694.ShaoY, DuanQ, QiuS. Adaptive analysis for phase-field model of brittle fracture of functionally graded materials. Eng Fract Mech. 2021;251(6):107783. doi:10.1016/j.engfracmech.2021.107783.ShaoY, DuanQ, ChenR. Adaptive meshfree method for fourth-order phase-field model of fracture using consistent integration schemes. Comput Mater Sci. 2024;233(1):112743. doi:10.1016/j.commatsci.2023.112743.LinQ, WangJ, HongJ, LiuZ, WangZ. A biomimetic generative optimization design for conductive heat transfer based on element-free Galerkin method. Int Commun Heat Mass Transf. 2019;100(17):67–72. doi:10.1016/j.icheatmasstransfer.2018.12.001.ZhangJ, ChenJ, PengJ, LiuT, WuS, ZhangH, et al.Periodic topology optimization of anisotropic heat conduction structure using the element-free Galerkin method. Eng Comput. 2025;41(5):3391–406. doi:10.1007/s00366-025-02153-9.ZhangJ, ZhaoL, DongJ, LiY, LiuB, ZhaoY, et al.Multi-material topology optimization of transient heat transfer structure based on element-free Galerkin method. Int Commun Heat Mass Transf. 2025;167(1):109383. doi:10.1016/j.icheatmasstransfer.2025.109383.ZhangJ, ZhaoL, ZhangH, LiuB, ChenT, HeX. Topology optimization of periodic heat transfer structure with anisotropic multi-material based on element-free Galerkin method. Structures. 2025;71:108107. doi:10.1016/j.istruc.2024.108107.ZhangJ, ZhangZ, WuS, ZhaoL, WuS, ZuoZ, et al.Topology optimization of multi-material structures with length-scale control based on Neural Style Transfer and element-free Galerkin method. Eur J Mech-A/Solids. 2025;112:105645. doi:10.1016/j.euromechsol.2025.105645.ZhengJ, YangX, LongS. Topology optimization with geometrically non-linear based on the element free Galerkin method. Int J Mech Mater Des. 2014;11(3):231–41. doi:10.1007/s10999-014-9257-y.ZhangY, GeW, ZhangY, ZhaoZ. Topology optimization method with direct coupled finite element-element-free Galerkin method. Adv Eng Softw. 2018;115(2):217–29. doi:10.1016/j.advengsoft.2017.09.012.UllahZ, UllahB, KhanW, ul IslamS. Proportional topology optimisation with maximum entropy-based meshless method for minimum compliance and stress constrained problems. Eng Comput. 2022;38(6):5541–61. doi:10.1007/s00366-022-01683-w.WangS, QianH, JuL. Topology optimization for minimizing the mean compliance under thermo-mechanical loads using element-free Galerkin method. Appl Math Model. 2024;136:115630. doi:10.1016/j.apm.2024.08.002.ZhangJ, LiuT, WangS, GongS, PengJ, ZuoQ. Thermomechanical coupling multi-objective topology optimization of anisotropic structures based on the element-free Galerkin method. Eng Optim. 2021;54(3):428–49. doi:10.1080/0305215x.2021.1872557.ZhangJ, PengJ, LiuT, ZhangH, ChenJ, LuoT, et al.Multi-objective periodic topology optimization of thermo-mechanical coupling structure with anisotropic materials by using the element-free Galerkin method. Int J Mech Mater Des. 2022;18(4):939–60. doi:10.1007/s10999-022-09600-1.WangS, YiW, QianH, JuL. Multi-objective topology optimization of thermoelastic structures based on points density using element-free Galerkin method. Eng Struct. 2025;326(10):119515. doi:10.1016/j.engstruct.2024.119515.LiX, ZhangS, WangY, ChenH. Analysis and application of the element-free Galerkin method for nonlinear sine-Gordon and generalized sinh-Gordon equations. Comput Mathem Appl. 2016;71(8):1655–78. doi:10.1016/j.camwa.2016.03.007.IlatiM. A meshless local moving Kriging method for solving Ginzburg-Landau equation on irregular domains. Eur Phys J Plus. 2020;135(11):873. doi:10.1140/epjp/s13360-020-00890-y.LiX, LiS. Analysis of an element-free Galerkin method for the nonlinear Schrödinger equation. Math Comput Simul. 2023;203:538–52. doi:10.1016/j.matcom.2022.06.031.LiX, CuiX, ZhangS. Analysis of a Crank-Nicolson fast element-free Galerkin method for the nonlinear complex Ginzburg-Landau equation. J Comput Appl Math. 2025;457(6):116323. doi:10.1016/j.cam.2024.116323.DehghanM, AbbaszadehM. Numerical study of three-dimensional Turing patterns using a meshless method based on moving Kriging element free Galerkin (EFG) approach. Comput Mathem Appl. 2016;72(3):427–54. doi:10.1016/j.camwa.2016.04.038.DehghanM, NarimaniN. The element-free Galerkin method based on moving least squares and moving Kriging approximations for solving two-dimensional tumor-induced angiogenesis model. Eng Comput. 2019;36(4):1517–37. doi:10.1007/s00366-019-00779-0.PathaniaT, EldhoTI, Bottacin-BusolinA. Coupled simulation of groundwater flow and multispecies reactive transport in an unconfined aquifer using the element-free Galerkin method. Eng Anal Bound Elem. 2020;121(3):31–49. doi:10.1016/j.enganabound.2020.08.019.SolinP, SegethK, DolezelI. Higher-order finite element methods. Boca Raton, FL, USA: Chapman and Hall/CRC; 2003.WangJ, SunF. A hybrid variational multiscale element-free galerkin method for convection-diffusion problems. Int J Appl Mech. 2019;11(7):1950063. doi:10.1142/s1758825119500637.ChengY, PengM, ChengY. A hybrid interpolating element-free Galerkin method for 3D steady-state convection diffusion problems. Appl Numer Math. 2025;208(6):21–37. doi:10.1016/j.apnum.2024.09.024.ZhangJ, YangY‐C, LiuF‐B, ChengH. Analyzing 3D steady variable coefficients convection-diffusion-reaction equations via a hybrid element-free galerkin method. Int J Numer Methods Fluids. 2025;97(6):996–1008. doi:10.1002/fld.5386.ZhangT, LiX. A variational multiscale interpolating element-free Galerkin method for convection-diffusion and Stokes problems. Eng Anal Bound Elem. 2017;82:185–93. doi:10.1016/j.enganabound.2017.06.013.JainS, BhargavaR. Natural convection flow on a bent wavy vertical enclosure filled with power-law nanofluid simulated by Element Free Galerkin method. Math Comput Simul. 2023;205:970–86. doi:10.1016/j.matcom.2022.10.033.XiangH, ZhangX. Variational multiscale element-free galerkin method and precise time step integration method for convection-diffusion problems. Numer Heat Trans Part A Appl. 2014;67(2):210–23. doi:10.1080/10407782.2014.923220.PeddavarapuS, RaghuramanS. Maximum entropy-based variational multiscale element-free Galerkin methods for scalar advection-diffusion problems. J Therm Anal Calorim. 2020;141(6):2527–40. doi:10.1007/s10973-020-09845-y.ZhangX, ZhangP, QinW, ShiX. An adaptive variational multiscale element free Galerkin method for convection-diffusion equations. Eng Comput. 2021;38(S4):3373–90. doi:10.1007/s00366-021-01469-6.ZhangX, XiangH. Variational multiscale element free Galerkin method for convection-diffusion-reaction equation with small diffusion. Eng Anal Bound Elem. 2014;46:85–92. doi:10.1016/j.enganabound.2014.05.010.LiX. A stabilized element-free Galerkin method for the advection-diffusion–reaction problem. Appl Math Lett. 2023;146:108831. doi:10.1016/j.aml.2023.108831.WangJ, WuY, XuY, SunF. A dimension-splitting variational multiscale element-free galerkin method for three-dimensional singularly perturbed convection-diffusion problems. Comput Model Eng Sci. 2023;135(1):341–56. doi:10.32604/cmes.2022.023140.DehghanM, AbbaszadehM. Variational multiscale element-free Galerkin method combined with the moving Kriging interpolation for solving some partial differential equations with discontinuous solutions. Comput Appl Math. 2017;37(3):3869–905. doi:10.1007/s40314-017-0546-6.BelytschkoT, OrganD, KrongauzY. A coupled finite element-element-free Galerkin method. Comput Mech. 1995;17(3):186–95. doi:10.1007/bf00364080.HuertaA, Fernández-MéndezS, LiuWK. A comparison of two formulations to blend finite elements and mesh-free methods. Comput Methods Appl Mech Eng. 2004;193(12–14):1105–17. doi:10.1016/j.cma.2003.12.009.HegenD. Element-free Galerkin methods in combination with finite element approaches. Comput Methods Appl Mech Eng. 1996;135(1–2):143–66. doi:10.1016/0045-7825(96)00994-2.WangH-P, WuC-T, GuoY, BotkinME. A coupled meshfree/finite element method for automotive crashworthiness simulations. Int J Impact Eng. 2009;36(10–11):1210–22. doi:10.1016/j.ijimpeng.2009.03.004.UllahZ, CoombsWM, AugardeCE. An adaptive finite element/meshless coupled method based on local maximum entropy shape functions for linear and nonlinear problems. Comput Methods Appl Mech Eng. 2013;267(13):111–32. doi:10.1016/j.cma.2013.07.018.UllahZ, CoombsW, AugardeC. Parallel computations in nonlinear solid mechanics using adaptive finite element and meshless methods. Eng Comput. 2016;33(4):1161–91. doi:10.1108/ec-06-2015-0166.ZhangX, HuZ, WangM. An adaptive interpolation element free Galerkin method based on a posteriori error estimation of FEM for Poisson equation. Eng Anal Bound Elem. 2021;130(1):186–95. doi:10.1016/j.enganabound.2021.05.020.ChengH, XingZ, LiuY. The improved element-free galerkin method for 3D steady convection-diffusion-reaction problems with variable coefficients. Mathematics. 2023;11(3):770. doi:10.3390/math11030770.LiX, WangQ. Analysis of the inherent instability of the interpolating moving least squares method when using improper polynomial bases. Eng Anal Bound Elem. 2016;73:21–34. doi:10.1016/j.enganabound.2016.08.012.WangJ-F, SunF-X, ChengY-M. An improved interpolating element-free Galerkin method with a nonsingular weight function for two-dimensional potential problems. Chin Phys B. 2012;21(9):090204. doi:10.1088/1674-1056/21/9/090204.ChengR, LiewKM. The reproducing kernel particle method for two-dimensional unsteady heat conduction problems. Comput Mech. 2009;45(1):1–10. doi:10.1007/s00466-009-0401-8.DehghanM, AbbaszadehM. Element free Galerkin approach based on the reproducing kernel particle method for solving 2D fractional Tricomi-type equation with Robin boundary condition. Comput Mathem Appl. 2017;73(6):1270–85. doi:10.1016/j.camwa.2016.11.020.GuL. Moving kriging interpolation and element-free Galerkin method. Int J Numer Methods Eng. 2002;56(1):1–11. doi:10.1002/nme.553.ZhengB, DaiB. A meshless local moving Kriging method for two-dimensional solids. Appl Math Comput. 2011;218(2):563–73. doi:10.1016/j.amc.2011.05.100.TuS, YangH, DongLL, HuangY. A stabilized moving Kriging interpolation method and its application in boundary node method. Eng Anal Bound Elem. 2019;100(1–4):14–23. doi:10.1016/j.enganabound.2017.12.016.WangF, LinG, ZhouY-H, ChenD-H. Element-free Galerkin scaled boundary method based on moving Kriging interpolation for steady heat conduction analysis. Eng Anal Bound Elem. 2019;106(3):440–51. doi:10.1016/j.enganabound.2019.05.027.SukumarN. Construction of polygonal interpolants: a maximum entropy approach. Int J Numer Methods Eng. 2004;61(12):2159–81. doi:10.1002/nme.1193.ArroyoM, OrtizM. Local maximum-entropy approximation schemes: a seamless bridge between finite elements and meshfree methods. Int J Numer Methods Eng. 2005;65(13):2167–202. doi:10.1002/nme.1534.SukumarN, WrightRW. Overview and construction of meshfree basis functions: from moving least squares to entropy approximants. Int J Numer Methods Eng. 2006;70(2):181–205. doi:10.1002/nme.1885.OrtizA, PusoMA, SukumarN. Maximum-entropy meshfree method for compressible and near-incompressible elasticity. Comput Methods Appl Mech Eng. 2010;199(25–28):1859–71. doi:10.1016/j.cma.2010.02.013.OrtizA, PusoMA, SukumarN. Maximum-entropy meshfree method for incompressible media problems. Finite Elem Anal Des. 2011;47(6):572–85. doi:10.1016/j.finel.2010.12.009.UllahZ, AugardeCE. Finite deformation elasto-plastic modelling using an adaptive meshless method. Comput Struct. 2013;118(16):39–52. doi:10.1016/j.compstruc.2012.04.001.UllahZ. Nonlinear solid mechanics analysis using the parallel selective element-free Galerkin method [Ph.D. thesis]. Durham, UK: Durham University, Department of Engineering; 2013.HuertaA, VidalY, VillonP. Pseudo-divergence-free element free Galerkin method for incompressible fluid flow. Comput Methods Appl Mech Eng. 2004;193(12–14):1119–36. doi:10.1016/j.cma.2003.12.010.RecioDP, Natal JorgeRM, DinisLMS. On the use of element-free Galerkin Method for problems involving incompressibility. Eng Anal Bound Elem. 2007;31(2):103–15. doi:10.1016/j.enganabound.2006.08.007.HuertaA, Fernández‐MéndezS. Time accurate consistently stabilized mesh-free methods for convection dominated problems. Int J Numer Methods Eng. 2003;56(9):1225–42. doi:10.1002/nme.602.VidalY, VillonP, HuertaA. Locking in the incompressible limit: pseudo-divergence-free element free Galerkin. Commun Numer Methods Eng. 2003;19(9):725–35. doi:10.1002/cnm.631.GuYT, LiuGR. Meshless techniques for convection dominated problems. Comput Mech. 2005;38(2):171–82. doi:10.1007/s00466-005-0736-8.LiX, ChenH, WangY. Error analysis in Sobolev spaces for the improved moving least-square approximation and the improved element-free Galerkin method. Appl Math Comput. 2015;262:56–78. doi:10.1016/j.amc.2015.04.002.LiX, LiS. On the stability of the moving least squares approximation and the element-free Galerkin method. Comput Mathem Appl. 2016;72(6):1515–31. doi:10.1016/j.camwa.2016.06.047.PantM, SharmaK. A comparative study of modeling material discontinuity using element free galerkin method. Procedia Eng. 2014;86:758–66. doi:10.1016/j.proeng.2014.11.095.ZhengH, LiW, DuX. Exact imposition of essential boundary condition and material interface continuity in Galerkin-based meshless methods. Int J Numer Methods Eng. 2016;110(7):637–60. doi:10.1002/nme.5370.AkhilSL, KrishnaIRP, AswathyM. Effect of non-dimensional length scale in element free Galerkin method for classical and strain driven nonlocal elasto-static problems. Comput Struct. 2025;312(5):107724. doi:10.1016/j.compstruc.2025.107724.ChenJ-S, WangH-P. New boundary condition treatments in meshfree computation of contact problems. Comput Methods Appl Mech Eng. 2000;187(3–4):441–68. doi:10.1016/s0045-7825(00)80004-3.RaoBN, RahmanS. An efficient meshless method for fracture analysis of cracks. Comput Mech. 2000;26(4):398–408. doi:10.1007/s004660000189.ChuY, WuJ, ChenP, ZhangC, WangD. A reproducing kernel gradient smoothing meshfree method with least squares stabilization for nearly incompressible elasticity. Eng Anal Bound Elem. 2026;183:106571. doi:10.1016/j.enganabound.2025.106571.HansboP. Nitsche’s method for interface problems in computational mechanics. GAMM-Mitteilungen. 2005;28(2):183–206. doi:10.1002/gamm.201490018.WuJ, WangD. An accuracy analysis of Galerkin meshfree methods accounting for numerical integration. Comput Methods Appl Mech Eng. 2021;375:113631. doi:10.1016/j.cma.2020.113631.BeisselS, BelytschkoT. Nodal integration of the element-free Galerkin method. Comput Methods Appl Mech Eng. 1996;139(1–4):49–74. doi:10.1016/s0045-7825(96)01079-1.HillmanM, ChenJ-S. An accelerated, convergent, and stable nodal integration in Galerkin meshfree methods for linear and nonlinear mechanics. Int J Numer Methods Eng. 2015;107(7):603–30. doi:10.1002/nme.5183.ChenJS, HuW, PusoMA, WuY, ZhangX. Strain smoothing for stabilization and regularization of galerkin meshfree methods. Berlin/Heidelberg, Germany: Springer Berlin Heidelberg; 2007. p. 57–75.ChenJ-S, WuC-T, YoonS, YouY. A stabilized conforming nodal integration for Galerkin mesh-free methods. Int J Numer Methods Eng. 2000;50(2):435–66. doi:10.1002/1097-0207(20010120)50:2<435::aid-nme32>3.0.co;2-a.DuanQ, LiX, ZhangH, BelytschkoT. Second-order accurate derivatives and integration schemes for meshfree methods. Int J Numer Methods Eng. 2012;92(4):399–424. doi:10.1002/nme.4359.DuanQ, LiX, ZhangH, WangB, GaoX. Quadratically consistent one-point (QC1) quadrature for meshfree Galerkin methods. Comput Methods Appl Mech Eng. 2012;245–246(1):256–72. doi:10.1016/j.cma.2012.07.019.GaoX, DuanQ, ShaoY, LiX, ChenB, ZhangH. Quadratically consistent one-point (QC1) integration for three-dimensional element-free Galerkin method. Finite Elem Anal Des. 2016;114:22–38. doi:10.1016/j.finel.2016.01.003.MaH, ChenJ, DengJ. Analysis of the dynamic response for Kirchhoff plates by the element-free Galerkin method. J Comput Appl Math. 2024;451(1):116093. doi:10.1016/j.cam.2024.116093.MaH, ChenJ, DengJ. Error analysis of the element-free Galerkin method for a nonlinear plate problem. Comput Mathem Appl. 2024;163(1):56–65. doi:10.1016/j.camwa.2024.03.020.WangD, ChenJ‐S. A Hermite reproducing kernel approximation for thin-plate analysis with sub-domain stabilized conforming integration. Int J Numer Methods Eng. 2007;74(3):368–90. doi:10.1002/nme.2175.PereiraMS, DonadonMV. Modified consistent element-free Galerkin method applied to Reissner-Mindlin plates. Thin-Walled Struct. 2025;212(2):113185. doi:10.1016/j.tws.2025.113185.LiX, DuanQ. Meshfree iterative stabilized Taylor-Galerkin and characteristic-based split (CBS) algorithms for incompressible N-S equations. Comput Methods Appl Mech Eng. 2006;195(44–47):6125–45. doi:10.1016/j.cma.2005.12.011.ZhangXH, OuyangJ, ZhangL. The characteristic-based split (CBS) meshfree method for free surface flow problems in ALE formulation. Int J Numer Methods Fluids. 2011;65(7):798–811. doi:10.1002/fld.2213.HughesTJR, FeijóoGR, MazzeiL, QuincyJ-B. The variational multiscale method—a paradigm for computational mechanics. Comput Methods Appl Mech Eng. 1998;166(1–2):3–24. doi:10.1016/s0045-7825(98)00079-6.ZhangT, LiX. A novel variational multiscale interpolating element-free Galerkin method for generalized Oseen problems. Comput Struct. 2018;209(1–3):14–29. doi:10.1016/j.compstruc.2018.08.002.FanY, ZhangX. The reduced variational multiscale element free Galerkin method for three-dimensional steady incompressible Stokes and Navier-Stokes equations. Eng Anal Bound Elem. 2024;169(1):105984. doi:10.1016/j.enganabound.2024.105984.ZhangT, LiX. Analysis of the element-free galerkin method with penalty for stokes problems. Entropy. 2022;24(8):1072. doi:10.1016/j.enganabound.2017.06.013.KamranianM, TatariM, DehghanM. Analysis of the stabilized element free Galerkin approximations to the Stokes equations. Appl Numer Math. 2020;150(10):325–40. doi:10.1016/j.apnum.2019.10.002.GohCM, NielsenPMF, NashMP. A stabilised mixed meshfree method for incompressible media: application to linear elasticity and Stokes flow. Comput Methods Appl Mech Eng. 2018;329:575–98. doi:10.1016/j.cma.2017.10.002.SamimiS, PakA. A fully coupled element-free Galerkin model for hydro-mechanical analysis of advancement of fluid-driven fractures in porous media. Int J Numer Anal Methods Geomech. 2016;40(16):2178–206. doi:10.1002/nag.2525.AdalatO, ScrimieriD. Efficient finite element mesh mapping using octree indexing. Cham, Switzerland: Springer Nature; 2024. p. 347–58.FlorianiLD, FellegaraR, MagilloP. Spatial indexing on tetrahedral meshes. In: Proceedings of the 18th SIGSPATIAL International Conference on Advances in Geographic Information Systems, GIS ’10. New York, NY, USA: ACM; 2010. p. 506–9.GhiaU, GhiaKN, ShinCT. High-Re solutions for incompressible flow using the Navier-Stokes equations and a multigrid method. J Comput Phys. 1982;48(3):387–411. doi:10.1016/0021-9991(82)90058-4.DingP. Solution of lid-driven cavity problems with an improved SIMPLE algorithm at high Reynolds numbers. Int J Heat Mass Transf. 2017;115(2):942–54. doi:10.1016/j.ijheatmasstransfer.2017.08.084.WahbaEM. Steady flow simulations inside a driven cavity up to Reynolds number 35,000. Comput Fluids. 2012;66:85–97. doi:10.1016/j.compfluid.2012.06.012.HachemE, RivauxB, KloczkoT, DigonnetH, CoupezT. Stabilized finite element method for incompressible flows with high Reynolds number. J Comput Phys. 2010;229(23):8643–65. doi:10.1016/j.jcp.2010.07.030.Al-AliHH, SelimMS. Simultaneous development of velocity and temperature profiles in the entrance region of a parallel plate channel: laminar flow with uniform wall heat flux. Chem Eng Commun. 1992;112(1):1–19. doi:10.1080/00986449208935988.ChengH, ZhangJ, XingZ. The hybrid element-free galerkin method for 3D helmholtz equations. Int J Appl Mech. 2022;14(9):2250084. doi:10.1142/s1758825122500843.SunF, WangJ, WuY, WeiQ. An improved element-free galerkin method based on the dimension splitting moving least-squares method for 2D potential problems in irregular domains. Int J Appl Mech. 2022;14(10):2250065. doi:10.1142/s175882512250065x.KrongauzY, BelytschkoT. Enforcement of essential boundary conditions in meshless approximations using finite elements. Comput Methods Appl Mech Eng. 1996;131(1–2):133–45. doi:10.1016/0045-7825(95)00954-x.RabczukT, XiaoSP, SauerM. Coupling of mesh-free methods with finite elements: basic concepts and test results. Commun Numer Methods Eng. 2006;22(10):1031–65. doi:10.1002/cnm.871.JohanssonA, KehletB, LarsonMG, LoggA. Multimesh finite element methods: solving PDEs on multiple intersecting meshes. Comput Methods Appl Mech Eng. 2019;343:672–89. doi:10.1016/j.cma.2018.09.009.StortiB, GarelliL, StortiM, D’ElíaJ. A matrix-free Chimera approach based on Dirichlet-Dirichlet coupling for domain composition purposes. Comput Mathem Appl. 2020;79(12):3310–30. doi:10.1016/j.camwa.2020.01.021.BrazellMJ, SitaramanJ, MavriplisDJ. An overset mesh approach for 3D mixed element high-order discretizations. J Comput Phys. 2016;322(3):33–51. doi:10.1016/j.jcp.2016.06.031.MengF, BanksJW, HenshawWD, SchwendemanDW. Fourth-order accurate fractional-step IMEX schemes for the incompressible Navier-Stokes equations on moving overlapping grids. Comput Methods Appl Mech Eng. 2020;366(1):113040. doi:10.1016/j.cma.2020.113040.ZhangZ, LiewKM, ChengY. Coupling of the improved element-free Galerkin and boundary element methods for two-dimensional elasticity problems. Eng Anal Bound Elem. 2008;32(2):100–7. doi:10.1016/j.enganabound.2007.06.006.MousaviH, AzhariM, SaadatpourMM, Sarrami-ForoushaniS. Application of improved element-free Galerkin combining with finite strip method for buckling analysis of channel-section beams with openings. Eng Comput. 2020;38(1):739–55. doi:10.1007/s00366-020-01087-8.HuangJ, Nguyen-ThanhN, LiW, ZhouK. An adaptive isogeometric-meshfree coupling approach for the limit analysis of cracked structures. Theor Appl Fract Mech. 2020;106(2):102426. doi:10.1016/j.tafmec.2019.102426.LuoT, ZhangJ, WuS, YinS, HeH, GongS. Steady heat transfer analysis for anisotropic structures using the coupled IGA-EFG method. Eng Anal Bound Elem. 2023;154:238–54. doi:10.1016/j.enganabound.2023.05.026.ChampagneO, PhamX-T. Numerical simulation of moving heat source in arc welding using the Element-free Galerkin method with experimental validation and numerical study. Int J Heat Mass Transf. 2020;154(4):119633. doi:10.1016/j.ijheatmasstransfer.2020.119633.MoarrefzadehA, JabbaripourB. Temperature distribution analysis and determination of residual stress due to welding using the element-free Galerkin method with experimental validation. Arch Appl Mech. 2025;95(8):195. doi:10.1007/s00419-025-02905-5.WuCT, HuW, WangH-P, LuH. A robust numerical procedure for the thermomechanical flow simulation of friction stir welding process using an adaptive element-free galerkin method. Math Probl Eng. 2015;2015(3):1–16. doi:10.1155/2015/486346.TalebiH, FröndM, dos SantosJF, KlusemannB. Thermomechanical simulation of friction stir welding of aluminum using an adaptive element-free galerkin method. PAMM. 2017;17(1):473–4. doi:10.1002/pamm.201710206.TalebiH, FroendM, KlusemannB. Application of adaptive element-free galerkin method to simulate friction stir welding of aluminum. Procedia Eng. 2017;207:580–5. doi:10.1016/j.proeng.2017.10.1024.ZhangJ, XuC, HuH, ZhangD, GongS, PengD, et al.Phase change heat transfer analysis of anisotropic structures with moving heat source using the element-free Galerkin method. Numer Heat Trans Part B Fund. 2023;84(4):392–414. doi:10.1080/10407790.2023.2208734.ChenS, DuanQ. An adaptive second-order element-free Galerkin method for additive manufacturing process. Comput Mater Sci. 2020;183(2):109911. doi:10.1016/j.commatsci.2020.109911.NaddafniaM, PakA, IranmaneshM, ToureiA. An element free galerkin simulation of CO2 sequestration in nonhomogeneous saline aquifers. Int J Environ Res. 2024;19(1):27. doi:10.1007/s41742-024-00692-5.TangL, YaoM, WangX, ZhangX. Non-uniform thermal behavior and shell growth within mould for wide and thick slab continuous casting. Steel Res Int. 2012;83(12):1203–13. doi:10.1002/srin.201200075.ZhangL, RongY-M, ShenH-F, HuangT-Y. Solidification modeling in continuous casting by finite point method. J Mater Process Technol. 2007;192-193(1–4):511–7. doi:10.1016/j.jmatprotec.2007.04.092.NguyenVP, RabczukT, BordasS, DuflotM. Meshless methods: a review and computer implementation aspects. Math Comput Simul. 2008;79(3):763–813. doi:10.1016/j.matcom.2008.01.003.Sai KumarS, MuthuN. Generalized displacement control technique for mode I/II fracture problems using cohesive zone model within a modified element-free Galerkin framework. Fat Fract Eng Mat Struct. 2023;46(11):4119–41. doi:10.1111/ffe.14118.GhorbaniM, NoormohammadiN, BoroomandB. Enrichment of the element free Galerkin method for cracks and notches without a priori knowledge of the analytical singularity order. Comput Mathem Appl. 2024;162:155–79. doi:10.1016/j.camwa.2024.03.007.ZhangJ, ChenT, ChenJ, PengX, ZhaoY, LuH, et al.Robust topology optimization of multi-material structure with load uncertainty based on element-free Galerkin method. Eng Struct. 2025;340:120786. doi:10.1016/j.engstruct.2025.120786.ZhangJ, ChenJ, GaoR, YangY, KuangW, WuS, et al.Multi-material topology optimization of negative thermal expansion metamaterial structures based on element-free Galerkin method. Comput Struct. 2026;321(9):108080. doi:10.1016/j.compstruc.2025.108080.ZhangJ, ZhangZ, ZhangH, WuS, WuS, ZuoZ, et al.Topology optimization of auxetic microstructures with isotropic and orthotropic multiple materials based on element-free Galerkin method. Eng Anal Bound Elem. 2024;166(1):105811. doi:10.1016/j.enganabound.2024.105811.VerdugoF, BadiaS. The software design of Gridap: a finite element package based on the Julia JIT compiler. Comput Phys Commun. 2022;276:108341. doi:10.1016/j.cpc.2022.108341.BadiaS, MartínAF, VerdugoF. GridapDistributed: a massively parallel finite element toolbox in Julia. J Open Source Soft. 2022;7(74):4157. doi:10.21105/joss.04157.